Introduction to Trigonometry Class 10 Previous Year Questions: 1 Mark (Chapter 8)

1-mark questions on Introduction to Trigonometry

This page has 41 1-mark questions on Introduction to Trigonometry from CBSE Class 10 board papers of 2023, 2024, 2025 and 2026. Of these, 24 appeared only in Standard papers and 17 appeared only in Basic papers. Question formats: 41 MCQs. 17 of them were asked in two or more paper sets.

0 of 41 practised

All 41 Introduction to Trigonometry 1-mark questions

Q1. Which of the following statements is false ?

Mark practised

(A) tan 45° = cot 45°
(B) sin 90° = tan 45°
(C) sin 30° = cos 30°
(D) sin 45° = cos 45°

Marks: 1
Level: Basic
Asked in: 6 paper sets
Years: 2025, 2026
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2025 Mathematics Basic, Set 430/1/3, Q15, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) sin 30° = cos 30°, because sin 30° and cos 30° have different values.
Why? From the table of trigonometric ratios of specific angles, sin 30° = 1/2 and cos 30° = √3/2; different values make the equality false.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/1/3: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing A because you think cot 45° is 0; tan 45° and cot 45° are both 1.
Choosing B because you think sin 90° is 0; sin 90° and tan 45° are both 1.
Marking C as true after swapping the values, taking sin 30° as √3/2 and cos 30° as 1/2.
Assuming sin A and cos A are equal for every angle because they are equal at 45°.

Why this works

Check each option against Table 8.1. Option A: tan 45° and cot 45° are equal. Option B: sin 90° and tan 45° are equal. Option C: sin 30° = 1/2, while cos 30° = √3/2, so the two are not equal. Option D: sin 45° and cos 45° are equal. So the false statement is C. Quick check: at 45°, a right isosceles triangle has equal opposite and adjacent sides, so its sine and cosine are equal; at 30° the opposite and adjacent sides are different, so sin 30° and cos 30° cannot be equal.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

Table 8.1 gives the exact values of the six trigonometric ratios for 0°, 30°, 45°, 60° and 90°. This solution reads the values for 30°, 45° and 90° from that table and compares the two sides of each option.

Important formulas

sin 30° = 1/2 · from Table 8.1
cos 30° = √3/2 · from Table 8.1
sin 45° = cos 45° = 1/√2 · from Table 8.1
tan 45° = 1 · from Table 8.1
cot 45° = 1 · from Table 8.1

Q2. The value of \(\left( \cot^2 A - \frac{1}{\sin^2 A} \right)\) is :

Mark practised

(A) more than 1
(B) 1
(C) 0
(D) -1

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2025, 2026
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2025 Mathematics Basic, Set 430/1/2, Q5, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) −1, because 1/sin²A = cosec²A, and cosec²A is 1 more than cot²A, so cot²A − cosec²A = −1.
Why? The identity cosec²A − cot²A = 1 shows the second term is exactly 1 more than the first; subtracting it leaves −1.

The full 1 mark is for the correct option.

Answer:

(D) −1

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/1/2: option (D).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Reversing the identity: thinking cot²A − cosec²A is 1, when it is −1.
Replacing 1/sin²A by sin²A instead of cosec²A; that loses the reciprocal and changes the value.

Why this works

Rewrite the second term: 1/sin²A = cosec²A. The identity cosec²A = 1 + cot²A is the key. Then cot²A − 1/sin²A = cot²A − cosec²A = cot²A − (1 + cot²A) = −1. The cot²A terms cancel, so the value is −1 for every angle for which the expression is defined. Quick check at A = 45°: cot²45° = 1 and 1/sin²45° = 2, so the expression is 1 − 2 = −1. Another correct method also earns full marks.

Key idea to remember

Trigonometric identity: see Key ideas used on this page.

Explanation

A trigonometric identity is an equation involving trigonometric ratios that is true for all values of the angle for which the ratios are defined. Here the identity cosec²A = 1 + cot²A shows that cosec²A exceeds cot²A by exactly 1, so cot²A − cosec²A = −1.

Important formulas

cosec A = 1/sin A · So 1/sin²A = cosec²A.
cosec²A − cot²A = 1 · True for 0° < A ≤ 90°.

Q3. \((\sin \theta + \cos \theta)^2 + (\sin \theta - \cos \theta)^2 =\)

Mark practised

(A) 1
(B) 2
(C) \(2 + 2 \sin \theta \cos \theta\)
(D) \(2 + 4 \sin \theta \cos \theta\)

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2025, 2026
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2025 Mathematics Basic, Set 430/2/2, Q5, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 2, because (sinθ+cosθ)² + (sinθ−cosθ)² = 2(sin²θ+cos²θ) = 2 × 1 = 2.
Why? The identity sin²θ+cos²θ = 1 is true for every angle θ; it replaces sin²θ+cos²θ by 1, and the cross terms cancel when the two squares are added.

The full 1 mark is for the correct option.

Answer:

(B) 2

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/2/2: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing option C by expanding (sinθ−cosθ)² as sin²θ+cos²θ = 1 and forgetting the −2sinθcosθ term, so the sum becomes (1+2sinθcosθ)+1 = 2+2sinθcosθ.
Choosing option D by taking the middle terms as adding instead of cancelling: (1+2sinθcosθ)+(1+2sinθcosθ) = 2+4sinθcosθ.
Choosing option A by replacing sin²θ+cos²θ with 1 and forgetting that the expression contains two such brackets.

Why this works

Expand the two squares: (sinθ+cosθ)² = sin²θ+2sinθcosθ+cos²θ and (sinθ−cosθ)² = sin²θ−2sinθcosθ+cos²θ. Adding them, the terms 2sinθcosθ and −2sinθcosθ cancel, leaving 2sin²θ+2cos²θ = 2(sin²θ+cos²θ). Since sin²θ+cos²θ = 1, the value is 2. Quick check with θ = 45°: sin45° = cos45° = 1/√2, so the first square becomes 2 and the second becomes 0; the sum is 2.

Key idea to remember

Trigonometric identity: sin²θ + cos²θ = 1: see Key ideas used on this page.

Explanation

An identity is an equation true for all values of the angle. Here sin²θ+cos²θ = 1 replaces the bracket by 1, so the whole expression becomes 2.

Important formulas

(a+b)² + (a−b)² = 2(a²+b²) · Combines the two squares in one step.
sin²θ + cos²θ = 1 · Basic trigonometric identity from Section 8.4.

Q4. \((\sec \theta - \cos \theta)^2 + \sin^2 \theta - \tan^2 \theta = ?\)

Mark practised

(A) 0
(B) 1
(C) 2
(D) 4

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2025, 2026
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2025 Mathematics Basic, Set 430/2/3, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 0, because (sec θ − cos θ)² + sin² θ − tan² θ = sec²θ − 2 + cos²θ + sin²θ − tan²θ = (sec²θ − tan²θ) + (sin²θ + cos²θ) − 2 = 1 + 1 − 2 = 0.
Why? Expanding (a−b)² gives a² − 2ab + b²; since sec θ and cos θ are reciprocals, 2 sec θ cos θ = 2, so the middle term is −2. The identities sin²θ + cos²θ = 1 and sec²θ − tan²θ = 1 then give 1 + 1 − 2 = 0.

The full 1 mark is for the correct option.

Answer:

(A) 0

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/2/3: option (A).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Expanding (sec θ − cos θ)² as sec²θ − cos²θ and missing the middle term −2 sec θ cos θ; the remaining expression is not constant, so none of the options matches.
Applying sec²θ − tan²θ = 1 but ignoring the sin²θ + cos²θ pair after expanding; this gives 1 − 2 = −1, which is not an option.

Why this works

Expand the first bracket: (sec θ − cos θ)² = sec²θ − 2 sec θ cos θ + cos²θ. Since sec θ = 1/cos θ, the cross term is 2 sec θ cos θ = 2. So the whole expression becomes sec²θ − tan²θ + sin²θ + cos²θ − 2. The identities sin²θ + cos²θ = 1 and sec²θ − tan²θ = 1 each give 1, so the value is 1 + 1 − 2 = 0. Quick check at θ = 45°: (√2 − 1/√2)² + (1/√2)² − 1² = 1/2 + 1/2 − 1 = 0.

Key idea to remember

Trigonometric Identities: see Key ideas used on this page.

Explanation

An equation involving trigonometric ratios of an angle is a trigonometric identity if it is true for all values of the angle for which it is defined. Here the expression is regrouped so that the two Pythagorean identities sin²θ + cos²θ = 1 and sec²θ − tan²θ = 1 can be applied; each reduces a part to 1, and the remaining −2 makes the total 0.

Important formulas

(a − b)² = a² − 2ab + b² · Algebraic identity used to expand (sec θ − cos θ)².
sin²θ + cos²θ = 1 · Pythagorean identity, true for all angles for which the ratios are defined.
sec²θ − tan²θ = 1 · Holds for 0° ≤ θ < 90°, where sec θ and tan θ are defined.
sec θ = 1/cos θ · Reciprocal relation that makes the cross term 2 sec θ cos θ equal to 2.

Q5. The value of \(\theta\) for which \(\sin 2\theta = \tan 45^\circ\) is :

Mark practised

(A) \(22.5^\circ\)
(B) \(30^\circ\)
(C) \(45^\circ\)
(D) \(90^\circ\)

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2025, 2026
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2025 Mathematics Basic, Set 430/3/3, Q13, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 45°, because tan 45° = 1 and sin 90° = 1, so 2θ = 90° and θ = 45°.
Why? Since tan 45° = 1 and sin 90° = 1, the equation sin 2θ = 1 forces 2θ = 90°, giving θ = 45°.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/3/3: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 90° because sin 90° = 1, without solving 2θ = 90° for θ.
Choosing 22.5° by halving 45° from tan 45°, instead of first writing sin 2θ = 1.

Q6. If sin θ = cos θ, (0° < θ < 90°), then value of (sec θ · sin θ) is :

Mark practised

(A) 1/√2
(B) √2
(C) 1
(D) 0

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2024
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2024 Mathematics Standard, Set 30/2/1, Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 1, because sin θ = cos θ gives θ = 45°, so sec θ · sin θ = √2 × 1/√2 = 1.
Why? At 45°, the sine and cosine ratios are equal. Because sec θ = 1/cos θ, the product sec θ · sin θ is tan θ, and tan 45° = 1.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/2/1: option (C).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing sin 45° itself, without multiplying by sec 45°.
Choosing sec 45° itself, without multiplying by sin 45°.
Thinking sin θ = cos θ means θ = 90°, where sec θ is not defined, instead of the acute angle 45°.

Why this works

For an acute angle, cos θ ≠ 0, so dividing sin θ = cos θ by cos θ gives tan θ = 1. The acute angle whose tangent is 1 is 45°. Also, sec θ · sin θ = (1/cos θ) sin θ = tan θ, so the required value is tan θ = 1. Quick check: at 45°, sin 45° and sec 45° are reciprocals, so their product is 1. Another correct method is to put the standard values directly.

Key idea to remember

Trigonometric ratios: see Key ideas used on this page.

Explanation

The trigonometric ratios of an acute angle are ratios of the sides of a right triangle. From the definitions, tan A = sin A / cos A and sec A = 1 / cos A. This solution uses those relations to simplify the required product before finding the angle.

Important formulas

tan A = sin A / cos A · Turns the product sec θ · sin θ into tan θ.
sec A = 1 / cos A · Definition of secant, used in the simplification.
tan 45° = 1 · From the standard values, this fixes θ = 45°.

Q7. If \(\sin 30^\circ \tan 45^\circ = \frac{\sec 60^\circ}{k}\) , then the value of k is :

Mark practised

(A) 4
(B) 3
(C) 2
(D) 1

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2025
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2025 Mathematics Standard, Set 30/2/1, Q13, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 4, because sin 30° tan 45° = (1/2)(1) = 1/2, sec 60° = 1/(1/2) = 2, and 1/2 = 2/k gives k = 4.
Why? The trigonometric ratios of specific angles have fixed standard values; substituting them converts the equation into a linear equation, and cross-multiplying gives k.

The full 1 mark is for the correct option.

Answer:

(A) 4

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/2/1: option (A).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 2 because sec 60° is 2, without dividing by the left side 1/2.
Thinking tan 45° is √2, or sec 60° is 1/√3, instead of using the standard values.
Reversing the cross-multiplication, which gives the reciprocal of the correct value.

Why this works

Using the standard values, sin 30° tan 45° = (1/2)(1) = 1/2 and sec 60° = 1/(1/2) = 2. So the equation becomes 1/2 = 2/k. Multiplying both sides by k gives k/2 = 2, hence k = 2 ÷ (1/2) = 4. Check: 2/4 = 1/2, which equals the left side. Another correct method also earns full marks.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

This topic gives the fixed values of sin, cos, tan, cosec, sec and cot for the angles 0°, 30°, 45°, 60° and 90°. This solution substitutes the values for 30°, 45° and 60° into the equation and solves the resulting linear equation for k.

Important formulas

sin 30° tan 45° = (1/2)(1) = 1/2 · Left side of the given equation.
sec 60° = 1/(1/2) = 2 · Since sec A = 1/cos A and cos 60° = 1/2.

Q8. If \(\sec\theta - \tan\theta = 2\), then \(\sec\theta + \tan\theta\) is equal to :

Mark practised

(A) \(\frac{1}{2}\)
(B) \(\sqrt{2}\)
(C) \(\frac{1}{\sqrt{2}}\)
(D) 2

Marks: 1
Level: Basic
Asked in: 3 paper sets
Years: 2025
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2025 Mathematics Basic, Set 430/4/2, Q2, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 1/2, because (sec θ − tan θ)(sec θ + tan θ) = sec²θ − tan²θ = 1, so 2(sec θ + tan θ) = 1, giving sec θ + tan θ = 1/2.
Why? By the identity sec²A − tan²A = 1, the product of the two factors equals 1.

The full 1 mark is for the correct option.

Answer:

(A) 1/2

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/4/2: option (A).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 2 just because the given difference is 2, without using the identity.
Thinking secθ + tanθ has the same value as secθ − tanθ; the plus sign changes the answer.
Finding θ from the given equation first, instead of using the identity directly.

Why this works

Use the identity sec²θ − tan²θ = 1. Factorise the left side as a difference of two squares: (secθ − tanθ)(secθ + tanθ) = 1. The question gives the first bracket as 2, so the second bracket must be 1 ÷ 2 = 1/2. Quick check: 2 × 1/2 = 1, which matches the identity. Another correct method also earns full marks.

Key idea to remember

Trigonometric identity: see Key ideas used on this page.

Explanation

A trigonometric identity is an equation involving trigonometric ratios that is true for all values of the angle for which the ratios are defined. The identity sec²A − tan²A = 1 comes from dividing Pythagoras theorem by the square of the adjacent side. Here it is used as a difference of squares to connect the given sum and difference.

Important formulas

sec²A − tan²A = 1 · Rearranged from 1 + tan²A = sec²A.
(secA − tanA)(secA + tanA) = sec²A − tan²A · Difference of squares.

Q9. The value of \(2 \sin^2 30^\circ + 3 \tan^2 60^\circ - \cos^2 45^\circ\) is :

Mark practised

(A) \(3\sqrt{3}\)
(B) \(\frac{19}{2}\)
(C) \(\frac{9}{4}\)
(D) 9

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2023
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2023 Mathematics Basic, Set 430/2/1, Q15, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 9, because 2 sin² 30° + 3 tan² 60° − cos² 45° = 2(1/2)² + 3(√3)² − (1/√2)² = 1/2 + 9 − 1/2 = 9.
Why? Substitute the specific-angle values for 30°, 60° and 45°, then simplify the arithmetic.

The full 1 mark is for the correct option.

Answer:

(D) 9

✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/2/1: option (D).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing 3 tan² 60° as 3√3 instead of 3(√3)², which is 9; this gives option A.
Forgetting to subtract cos² 45° after adding, which gives 19/2, option B.
Taking cos² 45° as 1 instead of (1/√2)², which is 1/2; the result is then not among the options.
Treating sin² 30° as sin 60°; it means (sin 30°)², which is 1/4.

Why this works

Put in the standard values: sin 30° as 1/2, tan 60° as √3, cos 45° as 1/√2. Then sin² 30° = (1/2)² = 1/4, tan² 60° = (√3)² = 3, cos² 45° = (1/√2)² = 1/2. Therefore 2 sin² 30° = 2 × 1/4 = 1/2 and 3 tan² 60° = 3 × 3 = 9. So the expression is 1/2 + 9 − 1/2 = 9. Option A, 3√3, comes from not squaring √3; option B, 19/2, comes from forgetting the final subtraction.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

This topic gives fixed values of the six trigonometric ratios for 0°, 30°, 45°, 60° and 90°. Here we need only three values: sin 30° is 1/2, tan 60° is √3 and cos 45° is 1/√2. The notation sin² 30° means (sin 30°)², so square each value before multiplying.

Important formulas

sin 30° is 1/2 · Standard value for 30°.
tan 60° is √3 · Square it to get 3.
cos 45° is 1/√2 · Square it to get 1/2.

Q10. \(9 \sec^2 A - 9 \tan^2 A\) is equal to :

Mark practised

(A) 9
(B) 0
(C) 8
(D) \(\frac{1}{9}\)

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2023
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2023 Mathematics Basic, Set 430/2/1, Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 9, because sec² A − tan² A = 1, so 9 sec² A − 9 tan² A = 9 × 1 = 9.
Why? The identity sec² A − tan² A = 1 is true for every angle A for which sec A and tan A are defined, so the bracket becomes 1 and 9 × 1 = 9.

The full 1 mark is for the correct option.

Answer:

(A) 9

✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/2/1: option (A).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 0 because you think sec² A and tan² A cancel each other, forgetting the factor 9 outside.
Choosing 8 because you subtract 1 from 9 instead of multiplying 9 by 1.
Choosing 1/9 because you confuse sec A with its reciprocal or take the reciprocal of 9.

Why this works

Take 9 common: 9 sec² A − 9 tan² A = 9(sec² A − tan² A). The identity sec² A − tan² A = 1 turns the bracket into 1, so the value is 9. Quick check with A = 60°: sec 60° = 2 and tan 60° = √3, so sec² 60° − tan² 60° = 4 − 3 = 1, and 9 × 1 = 9. Another correct method also earns full marks.

Key idea to remember

Trigonometric identity sec² A − tan² A = 1: see Key ideas used on this page.

Explanation

An identity is an equation true for all values of the angle for which the expressions are defined. In ΔABC right-angled at B, dividing AB² + BC² = AC² by AB² gives 1 + tan² A = sec² A, which is the same as sec² A − tan² A = 1. This solution factors out 9 and applies that identity.

Important formulas

sec² A − tan² A = 1 · True for 0° ≤ A < 90°; it makes the bracket equal to 1.

Q11. [3/4 tan² 30° - sec² 45° + sin² 60°] is equal to

Mark practised

(A) -1
(B) 5/6
(C) -3/2
(D) 1/6

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2023
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2023 Mathematics Standard, Set 30/1/1, Q9, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) -1, because (3/4)tan²30° - sec²45° + sin²60° = (3/4)(1/3) - 2 + 3/4 = 1/4 - 2 + 3/4 = -1.
Why? Substituting the standard values of the trigonometric ratios of specific angles turns the expression into simple arithmetic.

The full 1 mark is for the correct option.

Answer:

(A) -1

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/1/1: option (A).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using tan 30° as 1/2, which is sin 30°, instead of 1/√3.
Using sec 45° as 1/√2, which is cos 45°, instead of √2.
Reading sin²60° as 2 sin 60°; sin² A means the square of sin A, so the term is (√3/2)², not 2 × (√3/2).

Why this works

Put in the standard values of the specific angles. tan 30° = 1/√3, so tan²30° = (1/√3)² = 1/3. sec 45° = √2, so sec²45° = (√2)² = 2. sin 60° = √3/2, so sin²60° = (√3/2)² = 3/4. Then (3/4)tan²30° - sec²45° + sin²60° = (3/4)(1/3) - 2 + 3/4 = 1/4 - 2 + 3/4 = (1/4 + 3/4) - 2 = 1 - 2 = -1. So option A is correct.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

This topic gives fixed values of the trigonometric ratios for 0°, 30°, 45°, 60° and 90°. Replacing each ratio by its fixed value converts the given expression into an arithmetic expression, which is then simplified.

Important formulas

tan 30° = 1/√3 · Value for 30°.
sec 45° = √2 · Value for 45°.
sin 60° = √3/2 · Value for 60°.
sin² A = (sin A)² · Notation used for powers of trigonometric ratios.

Q12. \(\frac{\cos^2 \theta}{\sin^2 \theta} - \frac{1}{\sin^2 \theta}\), in simplified form, is:

Mark practised

(A) \(\tan^2 \theta\)
(B) \(\sec^2 \theta\)
(C) 1
(D) -1

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2023
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2023 Mathematics Standard, Set 30/6/1, Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) -1, because (cos²θ − 1)/sin²θ = (cos²θ − (cos²θ + sin²θ))/sin²θ = −sin²θ/sin²θ = −1.
Why? By the trigonometric identity cos²θ + sin²θ = 1, we replace 1 with cos²θ + sin²θ, so cos²θ − 1 = −sin²θ and the fraction reduces to −1.

The full 1 mark is for the correct option.

Answer:

(D) -1

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/6/1: option (D).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing the combined numerator as cos²θ − 1 and then replacing it with sin²θ, forgetting that cos²θ − 1 = −sin²θ.
Simplifying cos²θ/sin²θ to tan²θ, although cos²θ/sin²θ = cot²θ.
Cancelling sin²θ but keeping the positive sign, giving the positive result 1 instead of −1.

Why this works

Put the two terms over one denominator: (cos²θ − 1)/sin²θ. The identity cos²θ + sin²θ = 1 gives 1 = cos²θ + sin²θ, so cos²θ − 1 = −sin²θ. Hence the expression equals −sin²θ/sin²θ = −1. This is valid wherever the expression is defined, that is, where sin θ ≠ 0. Quick check with θ = 60°: cos²60° = (1/2)² = 1/4 and sin²60° = (√3/2)² = 3/4, so (1/4 − 1)/(3/4) = (−3/4)/(3/4) = −1. Another correct method also earns full marks.

Key idea to remember

Trigonometric identity: see Key ideas used on this page.

Explanation

A trigonometric identity is an equation involving trigonometric ratios that is true for all values of the angle for which it is defined. This solution uses cos²θ + sin²θ = 1 to replace 1, then cancels sin²θ to get −1.

Important formulas

cos²θ + sin²θ = 1 · Use it to rewrite 1 as cos²θ + sin²θ in the numerator.

Q13. If \(\cos \theta = \frac{\sqrt{3}}{2}\) and \(\sin \phi = \frac{1}{2}\) , then \(\tan (\theta + \phi)\) is :

Mark practised

(A) \(\sqrt{3}\)
(B) \(\frac{1}{\sqrt{3}}\)
(C) 1
(D) not defined

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2024
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2024 Mathematics Standard, Set 30/3/1, Q8, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) √3, because cos 30° = √3/2 and sin 30° = 1/2, so θ = 30°, φ = 30°, θ + φ = 60° and tan 60° = √3.
Why? A trigonometric ratio value from the table of specific angles identifies its angle; once θ and φ are known, the tangent of their sum is read from the same table.

The full 1 mark is for the correct option.

Answer:

(A) √3

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/3/1: option (A).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing D because θ is taken as 60° (confusing cos 30° = √3/2 with sin 60° = √3/2) and φ as 30°, so the sum is taken as 90° and tan 90° is not defined.
Choosing B because tan θ = tan 30° = 1/√3, without adding φ first.

Why this works

Compare the given ratios with the table of trigonometric ratios of specific angles. cos 30° = √3/2, so θ = 30°. sin 30° = 1/2, so φ = 30°. Therefore θ + φ = 60°, and the table gives tan 60° = √3. So the correct option is (A). Option D is for when the tangent is not defined, which does not happen here.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

The table of trigonometric ratios of specific angles gives the values for 0°, 30°, 45°, 60° and 90°. Here cos θ = √3/2 matches cos 30° and sin φ = 1/2 matches sin 30°, so both angles are 30°. Their sum is 60°, and the table gives tan 60° = √3.

Important formulas

cos 30° = √3/2 · used to find θ from the given cos θ
sin 30° = 1/2 · used to find φ from the given sin φ
tan 60° = √3 · value of tan for the sum θ + φ

Q14. If \(\sin \theta = 1\), then the value of \(\frac{1}{2} \sin \left( \frac{\theta}{2} \right)\) is :

Mark practised

(A) \(\frac{1}{2\sqrt{2}}\)
(B) \(\frac{1}{\sqrt{2}}\)
(C) \(\frac{1}{2}\)
(D) 0

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2024
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2024 Mathematics Standard, Set 30/4/1, Q10, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 1/(2√2), because sin θ = 1, so θ is 90°, θ/2 is 45°, and (1/2) sin 45° = (1/2)(1/√2) = 1/(2√2).
Why? The specific-angles table fixes sin 90° as 1, so the angle whose sine is 1 is 90°; the same table gives sin 45° as 1/√2, and the outer factor 1/2 is then applied.

The full 1 mark is for the correct option.

Answer:

(A) 1/(2√2)

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/4/1: option (A).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 1/√2 after finding sin 45°, because the factor 1/2 outside the sine still has to be multiplied.
Thinking the half applies after the sine, so θ/2 is treated as 90° instead of 45°.
Taking θ as 0° and getting 0, because the angle whose sine is 1 is 90°, not 0°.

Why this works

The specific-angles table gives sin 90° = 1. Since sin θ = 1, θ is 90°. Then θ/2 is 45°, and the table gives sin 45° = 1/√2. So (1/2) sin 45° = (1/2)(1/√2) = 1/(2√2). The other options miss this: 1/√2 is sin 45° before the outer half is applied, 1/2 is not sin 45°, and 0 is the sine of 0°.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

The specific-angles table in the chapter gives fixed values of the six trigonometric ratios for 0°, 30°, 45°, 60° and 90°. This solution uses the table's sine value at 90° and at 45° and substitutes them into the given expression.

Important formulas

sin 90° · Value from the specific-angles table is 1.
sin 45° · Value from the specific-angles table is 1/√2.

Q15. If \(4 \sec \theta - 5 = 0\) , then the value of \(\cot \theta\) is :

Mark practised

(A) \(\frac{3}{4}\)
(B) \(\frac{4}{5}\)
(C) \(\frac{5}{3}\)
(D) \(\frac{4}{3}\)

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2024
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2024 Mathematics Standard, Set 30/5/1, Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 4/3, because sec θ = 5/4, so cos θ = 4/5 and sin θ = √(1 − (4/5)²) = 3/5, hence cot θ = (4/5) ÷ (3/5) = 4/3.
Why? sec θ is the reciprocal of cos θ, and cot θ = cos θ/sin θ; the identity sin² θ + cos² θ = 1 gives sin θ when cos θ is known.

The full 1 mark is for the correct option.

Answer:

(D) 4/3

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/5/1: option (D).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Picking 3/4 (option A): that is tan θ, not cot θ.
Picking 4/5 (option B): that is cos θ, not cot θ.
Picking 5/3 (option C): that is cosec θ, not cot θ.

Why this works

From 4 sec θ − 5 = 0, we get sec θ = 5/4. Since sec θ is the reciprocal of cos θ, cos θ = 4/5. In the right triangle, sec θ = hypotenuse/adjacent, so take hypotenuse = 5k and adjacent = 4k. By Pythagoras theorem, opposite = √((5k)² − (4k)²) = 3k. Therefore sin θ = opposite/hypotenuse = 3/5, and cot θ = adjacent/opposite = 4/3. You can check: 4 × (5/4) − 5 = 0. For an acute angle, all these ratios are positive.

Key idea to remember

Trigonometric ratios of an angle: see Key ideas used on this page.

Explanation

The trigonometric ratios are defined using the sides of a right triangle. If one ratio is known, the remaining ratios can be determined. Here sec θ gives the hypotenuse and the adjacent side in the ratio 5 : 4; the opposite side is then found by Pythagoras theorem, and cot θ is adjacent divided by opposite.

Important formulas

sec θ = 1/cos θ · Secant is the reciprocal of cosine.
cot θ = cos θ/sin θ · Cotangent is the reciprocal of tangent.
sin² θ + cos² θ = 1 · Used to find sin θ when cos θ is known.

Q16. If \(\tan 3\theta = \sqrt{3}\), then \(\frac{\theta}{2}\) equals :

Mark practised

(A) 60°
(B) 30°
(C) 20°
(D) 10°

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2025
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2025 Mathematics Standard, Set 30/3/1, Q1, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 10°, because tan 60° = √3, so 3θ = 60°, θ = 20° and θ/2 = 10°.
Why? The trigonometric ratio of the specific angle 60° is tan 60° = √3, so the equation tan 3θ = √3 gives 3θ = 60°.

The full 1 mark is for the correct option.

Answer:

(D) 10°

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/3/1: option (D).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 20° because 3θ = 60° gives θ = 20°, but the question asks for θ/2, which is 10°.
Choosing 60° because tan 60° = √3, but 60° is the value of 3θ, not θ/2.
Using tan 30° = 1/√3 and concluding 3θ = 30°, which gives θ/2 = 5°; the correct standard value is tan 60° = √3.

Why this works

Compare tan 3θ with the standard value tan 60° = √3. This makes 3θ = 60°. Dividing by 3 gives θ = 20°, and dividing by 2 gives θ/2 = 10°. Alternatively, divide 3θ = 60° by 6 directly to get θ/2 = 10°. Check: if θ/2 = 10°, then θ = 20° and 3θ = 60°, and tan 60° = √3.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

The NCERT standard values give tan 60° = √3. When a trigonometric ratio equals one of these standard values, the corresponding angle can be read directly. Here tan 3θ = √3 means 3θ is the angle whose tangent is √3, so 3θ = 60°. The options are all small positive angles, so the acute angle 60° is the intended one. Then θ = 20° and θ/2 = 10°.

Important formulas

tan 60° = √3 · Standard value of the tangent of 60°.
θ/2 = (3θ)/6 · After finding 3θ = 60°, dividing by 6 gives θ/2 directly.

Q17. Given that \(\sin \theta = \frac{a}{b}\) , then \(\cos \theta\) is equal to :

Mark practised

(A) \(\frac{b}{\sqrt{b^2 - a^2}}\)
(B) \(\frac{b}{a}\)
(C) \(\frac{\sqrt{b^2 - a^2}}{b}\)
(D) \(\frac{a}{\sqrt{b^2 - a^2}}\)

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2026
Topic: 8.2 Trigonometric Ratios
Source: (CBSE 2026 Mathematics Standard, Set 30/1/1, Q9, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) \(\frac{\sqrt{b^2 - a^2}}{b}\), because \(\sin \theta = \frac{a}{b}\) gives opposite \(= a\) and hypotenuse \(= b\), so adjacent \(= \sqrt{b^2 - a^2}\) and \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\).
Why? By the definitions of sine and cosine in a right triangle, the third side is found by Pythagoras theorem, and cosine is adjacent over hypotenuse.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/1/1: option (C).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing b/a, which is cosec θ, the reciprocal of sin θ, not cos θ.
Choosing b/√(b²−a²), which is sec θ, not cos θ.
Choosing a/√(b²−a²), which is tan θ, not cos θ.
Writing just √(b²−a²) and forgetting to divide by the hypotenuse b.

Why this works

Let θ be an acute angle of a right triangle. By definition, sin θ = opposite/hypotenuse. Since sin θ = a/b, take opposite = a and hypotenuse = b; any common factor cancels. By Pythagoras theorem, adjacent = √(b²−a²). So cos θ = adjacent/hypotenuse = √(b²−a²)/b, which is option C. Quick check with a 3-4-5 triangle: sin θ = 3/5 and the formula gives √(5²−3²)/5 = 4/5, and 3² + 4² = 5².

Key idea to remember

Trigonometric ratios: see Key ideas used on this page.

Explanation

NCERT defines the trigonometric ratios of an acute angle as ratios of the sides of a right triangle. When one ratio is known, the remaining ratios can be determined by using Pythagoras theorem to find the missing side. NCERT Example 1 uses the same idea: given tan A = 4/3, it finds sin A and cos A.

Important formulas

sin θ = opposite/hypotenuse · Definition of sine.
cos θ = adjacent/hypotenuse · Definition of cosine.
(hypotenuse)² = (opposite)² + (adjacent)² · Pythagoras theorem in a right triangle.

Q18. \(\cot^2 \theta - \frac{1}{\sin^2 \theta}\) is equal to :

Mark practised

(A) 1
(B) 2
(C) -2
(D) -1

Marks: 1
Level: Standard
Compartment: 2023
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2023 Compartment Mathematics Standard, Set 30/C/1, Q6, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) -1, because 1/sin²θ = cosec²θ and cosec²θ - cot²θ = 1, so cot²θ - cosec²θ = -1.
Why? By the identity connecting cosec²A and cot²A, reversing the order of the two terms changes the sign, and 1/sin²θ is cosec²θ.

The full 1 mark is for the correct option.

Answer:

(D) -1

✓ Answer matches the CBSE answer key, 2023 Compartment, Paper 30/C/1: option (D).

Open CBSE’s official marking scheme for the 2023 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 1, which is the value of cosec²θ - cot²θ, because the given expression has the two terms in the opposite order.
Stopping after rewriting 1/sin²θ as cosec²θ without using the identity cosec²θ = 1 + cot²θ, so the constant -1 is never reached.
Using the identity sec²θ = 1 + tan²θ instead, because the expression involves cot θ and 1/sin²θ, not tan θ and 1/cos²θ.

Why this works

Since cosec θ is the reciprocal of sin θ, 1/sin²θ = cosec²θ, so the expression becomes cot²θ - cosec²θ. The identity cosec²A = 1 + cot²A is the same as cosec²A - cot²A = 1. Therefore cot²θ - cosec²θ is the negative of that, which is -1. For instance, when the angle is 45°, cot²45° = 1 and 1/sin²45° = 2, so the expression is 1 - 2 = -1.

Key idea to remember

Trigonometric identity: see Key ideas used on this page.

Explanation

An equation involving trigonometric ratios of an angle is a trigonometric identity when it is true for all values of the angle for which the expressions are defined. NCERT derives cosec²A = 1 + cot²A by dividing AB² + BC² = AC² by BC². This solution rewrites 1/sin²θ as cosec²θ and applies that identity to get the constant -1.

Important formulas

cosec θ = 1/sin θ · Definition of cosecant, so 1/sin²θ = cosec²θ.
cosec² A = 1 + cot² A · Trigonometric identity; it gives the constant -1 when cot²A - cosec²A is formed.

Q19. \(2 \cos^2 \theta (1 + \tan^2 \theta)\) is equal to :

Mark practised

(A) 0
(B) 1
(C) 2
(D) 3

Marks: 1
Level: Standard
Compartment: 2023
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2023 Compartment Mathematics Standard, Set 30/C/2, Q6, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 2, because 1 + tan² θ = sec² θ = 1/cos² θ, so 2 cos² θ × 1/cos² θ = 2.
Why? The identity 1 + tan² θ = sec² θ and the reciprocal relation sec θ = 1/cos θ together cancel the cos² θ, leaving the constant 2.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2023 Compartment, Paper 30/C/2: option (C).

Open CBSE’s official marking scheme for the 2023 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Cancelling cos² θ with sec² θ but forgetting the 2 outside, so choosing 1 instead of 2.
Writing 1 + tan² θ as sec θ without the square, which leaves an extra cos θ and gives 2 cos θ.
Testing at 90°, where tan θ is not defined, and wrongly concluding the expression has no value.

Why this works

Replace the bracket by the identity 1 + tan² θ = sec² θ. Because sec θ = 1/cos θ, the product becomes 2 cos² θ × 1/cos² θ = 2, whatever θ is (where the ratios are defined). Quick check: at 45°, cos²45° = 1/2 and 1 + tan²45° = 2, so the expression is 2 × (1/2) × 2 = 2, the same as option (C).

Key idea to remember

Trigonometric identity: see Key ideas used on this page.

Explanation

An equation involving trigonometric ratios of an angle is called a trigonometric identity if it is true for all values of the angle(s) involved. Here the identity 1 + tan² θ = sec² θ is used to replace the bracket, and then sec θ = 1/cos θ simplifies the whole product to 2.

Important formulas

1 + tan² θ = sec² θ · Standard trigonometric identity used to replace the bracket.
sec θ = 1/cos θ · Reciprocal relation; hence sec² θ = 1/cos² θ.
cos² θ × sec² θ = 1 · Direct consequence of the reciprocal relation, giving the cancellation.

Q20. 8 (cos² A + sin² A) is equal to :

Mark practised

(A) 1
(B) 0
(C) 9
(D) 8

Marks: 1
Level: Basic
Years: 2023
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2023 Mathematics Basic, Set 430/2/2, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 8, because cos² A + sin² A = 1, so 8(cos² A + sin² A) = 8 × 1 = 8.
Why? The trigonometric identity cos² A + sin² A = 1 is true for all angles A, so the bracket is always 1; multiplying the bracket by 8 gives the value of the expression.

The full 1 mark is for the correct option.

Answer:

(D) 8

✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/2/2: option (D).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing (A) 1, which is the value of the bracket cos² A + sin² A alone, before multiplying by 8.
Choosing (C) 9 by adding 8 and 1 instead of multiplying; the 8 multiplies the whole bracket.
Choosing (B) 0 by thinking the bracket could be 0; the identity keeps the bracket at 1 for every angle.

Why this works

In ΔABC right-angled at B, AB² + BC² = AC². Dividing every term by AC² gives (AB/AC)² + (BC/AC)² = 1, that is cos² A + sin² A = 1. This identity is true for every angle A, so the bracket is always 1 and 8(cos² A + sin² A) = 8 × 1 = 8. Quick check with A = 0°: sin 0° = 0 and cos 0° = 1, so sin² 0° + cos² 0° = 0 + 1 = 1, and the product is 8. Option (A) 1 is the bracket alone without the factor 8; option (C) 9 comes from adding 8 and 1 instead of multiplying; option (B) 0 would need the bracket to be 0, which never happens.

Key idea to remember

Trigonometric identity: see Key ideas used on this page.

Explanation

An equation involving trigonometric ratios of an angle is called a trigonometric identity if it is true for all values of the angle. The chapter proves cos² A + sin² A = 1 by dividing the Pythagoras relation by AC². This solution applies that identity, so the bracket becomes 1 and the whole expression becomes 8.

Important formulas

cos² A + sin² A = 1 · This identity is true for all A with 0° ≤ A ≤ 90°.

Q21. The value of \(5 \sin^2 90^\circ - 2 \cos^2 0^\circ\) is :

Mark practised

(A) \(-2\)
(B) \(5\)
(C) \(3\)
(D) \(-3\)

Marks: 1
Level: Basic
Years: 2023
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2023 Mathematics Basic, Set 430/4/1, Q10, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 3, because sin 90° = 1 and cos 0° = 1, so 5(1)² − 2(1)² = 3.
Why? For 90°, the side opposite the angle equals the hypotenuse, so sin 90° = 1; for 0°, the side adjacent to the angle equals the hypotenuse, so cos 0° = 1. Substituting these standard values gives the value of the expression.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/4/1: option (C).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing sin 90° as 0 or cos 0° as 0.
Taking sin² 90° as 2 × sin 90°, instead of (sin 90°)².
Taking cos² 0° as 2 × cos 0° and getting 1 instead of 3.

Why this works

This question only needs the standard values of sin 90° and cos 0°. From the specific-angle values, sin 90° = 1 and cos 0° = 1. Substituting gives 5 × 1² − 2 × 1² = 5 − 2 = 3, so the correct choice is (C) 3. A quick check: the whole expression reduces to 5 − 2, and 5 − 2 = 3.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

For the specific angles 0°, 30°, 45°, 60° and 90°, the trigonometric ratios have fixed standard values. This solution uses sin 90° = 1 and cos 0° = 1, and then evaluates the given expression by substitution.

Important formulas

sin 90° = 1 · Standard value used for substitution.
cos 0° = 1 · Standard value used for substitution.
sin² A means (sin A)² · The square is applied to the value of sin A, not to the angle.

Q22. \(\left[ \frac{5}{8} \sec^2 60^\circ - \tan^2 60^\circ + \cos^2 45^\circ \right]\) is equal to

Mark practised

(A) \(\frac{-5}{3}\)
(B) \(\frac{-1}{2}\)
(C) 0
(D) \(\frac{-1}{4}\)

Marks: 1
Level: Standard
Years: 2023
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2023 Mathematics Standard, Set 30/1/2, Q17, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 0, because 5/8 × sec² 60° − tan² 60° + cos² 45° = 5/8 × 4 − 3 + 1/2 = 0.
Why? The values of sec 60°, tan 60° and cos 45° are fixed standard values; substituting them and simplifying gives the value of the expression.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/1/2: option (C).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing sec 60° as 1/2, which is cos 60°, instead of 2.
Leaving out the + cos² 45° term; the first two terms alone give −1/2, which is option B.
Writing tan² 60° as tan 120° instead of squaring tan 60°.

Why this works

From the standard values for specific angles, cos 60° = 1/2, so sec 60° = 1/(1/2) = 2 and sec² 60° = 4. Also tan 60° = √3, so tan² 60° = (√3)² = 3. And cos 45° = 1/√2, so cos² 45° = 1/2. Now 5/8 × 4 = 5/2, so the expression is 5/2 − 3 + 1/2 = 0. Hence the expression equals 0, which is option (C).

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

The NCERT section 8.3 gives the values of the trigonometric ratios for 0°, 30°, 45°, 60° and 90°. Substituting those values for 60° and 45° turns the expression into an ordinary arithmetic calculation.

Important formulas

sec 60° = 2, tan 60° = √3, cos 45° = 1/√2 · Standard values from the table in NCERT Section 8.3.
sec² A − tan² A = 1 · Gives tan² 60° = sec² 60° − 1 = 4 − 1 = 3.

Q23. If \(\sec \theta - \tan \theta = \frac{1}{3}\), then the value of \((\sec \theta + \tan \theta)\) is :

Mark practised

(A) \(\frac{4}{3}\)
(B) \(\frac{2}{3}\)
(C) \(\frac{1}{3}\)
(D) 3

Marks: 1
Level: Standard
Years: 2023
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2023 Mathematics Standard, Set 30/5/3, Q9, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 3, because the identity sec²θ − tan²θ = 1 gives (secθ − tanθ)(secθ + tanθ) = 1, so 1 ÷ (1/3) = 3.
Why? By the trigonometric identity 1 + tan²θ = sec²θ, the product of the two factors is 1, so the second factor is the reciprocal of the first.

The full 1 mark is for the correct option.

Answer:

(D) 3

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/5/3: option (D).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 1/3 because secθ + tanθ looks the same as secθ − tanθ, without using the identity.
Taking sec²θ − tan²θ to be 0, which makes the product 0 instead of 1.

Why this works

Use the identity 1 + tan²θ = sec²θ, which is true for 0° ≤ θ < 90°. Rewrite it as sec²θ − tan²θ = 1. The left side is a difference of squares, so (secθ − tanθ)(secθ + tanθ) = 1. The question gives secθ − tanθ = 1/3. Therefore secθ + tanθ = 1 ÷ (1/3) = 3. Quick check: if secθ + tanθ = 3 and secθ − tanθ = 1/3, then adding gives 2secθ = 10/3, so secθ = 5/3; subtracting gives 2tanθ = 8/3, so tanθ = 4/3. Then sec²θ − tan²θ = 25/9 − 16/9 = 1, which matches the identity.

Key idea to remember

Trigonometric identity: see Key ideas used on this page.

Explanation

A trigonometric identity is an equation involving trigonometric ratios that is true for all values of the angle for which the ratios are defined. Here 1 + tan²θ = sec²θ, so sec²θ − tan²θ = 1. Factorising the left side as a difference of squares lets us use the given value directly.

Important formulas

sec²θ − tan²θ = 1 · Equivalent to 1 + tan²θ = sec²θ, true for 0° ≤ θ < 90°.
(a − b)(a + b) = a² − b² · Factorises the left side into two factors.

Q24. The maximum value of \(\frac{1}{\operatorname{cosec} \theta}\), (0 ≤ θ ≤ 90°) is :

Mark practised

(A) \(\frac{\sqrt{3}}{2}\)
(B) 1
(C) \(\frac{1}{\sqrt{2}}\)
(D) \(\sqrt{2}\)

Marks: 1
Level: Basic
Compartment: 2024
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2024 Compartment Mathematics Basic, Set 430(B), Q12, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 1, because 1/cosec θ = sin θ, whose greatest value for 0° ≤ θ ≤ 90° is 1.
Why? cosec θ is defined as 1/sin θ, so 1/cosec θ equals sin θ; over 0° to 90°, sin θ rises from 0 to 1, so its greatest value is 1.

The full 1 mark is for the correct option.

Answer:

(B) 1

✓ Answer matches the CBSE answer key, 2024 Compartment, Paper 430(B): option (B).

Open CBSE’s official marking scheme for the 2024 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing √3/2 because it is sin 60°, without checking all angles from 0° to 90°.
Picking 1/√2 because it is sin 45°, instead of first simplifying 1/cosec θ to sin θ.
Thinking 1/cosec θ is cosec θ itself, and then choosing √2.

Why this works

First simplify the expression: 1/cosec θ = sin θ, because cosec θ = 1/sin θ. On the interval 0° to 90°, the sine ratio increases from 0 to 1, so its greatest value is 1, reached at θ = 90°. A quick check: √3/2 is sin 60°, 1/√2 is sin 45°, and √2 is larger than 1, which is impossible for a sine value in this interval.

Key idea to remember

Cosecant as the reciprocal of sine: see Key ideas used on this page.

Explanation

The NCERT defines cosec A = 1/sin A. Therefore 1/cosec θ = sin θ. To find the maximum of a trigonometric ratio over a given range, look at how the ratio behaves in that range; for 0° ≤ θ ≤ 90°, sin θ increases from 0 to 1, so the maximum value is 1.

Important formulas

cosec θ = 1/sin θ · Definition of cosecant: reciprocal of sine.
1/cosec θ = sin θ · Reciprocal of cosecant is sine.
sin 90° = 1 · Greatest value of sine in 0° to 90°.

Q25. For what value of \(\theta\), \(\sin^2 \theta + \sin \theta + \cos^2 \theta\) is equal to 2 ?

Mark practised

(A) \(45^\circ\)
(B) \(0^\circ\)
(C) \(90^\circ\)
(D) \(30^\circ\)

Marks: 1
Level: Basic
Years: 2024
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2024 Mathematics Basic, Set 430/1/2, Q13, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 90°, because at θ = 90°, sin²90° + cos²90° = 1² + 0² = 1, so the expression becomes 1 + 1 = 2.
Why? The identity sin²θ + cos²θ = 1 is true for all θ, so the expression equals 1 + sin θ. Among the options, sin θ = 1 only at θ = 90°.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/1/2: option (C).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Testing each option by substituting into sin²θ + sin θ + cos²θ without first using the identity sin²θ + cos²θ = 1.
Choosing 0° because sin 0° is 0 and cos²0° = 1, without adding them, so the total is 1, not 2.
Thinking that sin θ alone equals 2, and then wondering why no option works.

Why this works

The identity sin²θ + cos²θ = 1 is true for every angle. So the given expression is 1 + sin θ. The question becomes: for which option is 1 + sin θ = 2, that is, sin θ = 1? Among 0°, 30°, 45° and 90°, sin θ = 1 only at 90°. Quick check: at 90°, sin²90° + sin 90° + cos²90° = 1² + 1 + 0² = 2. At 0°, sin²0° + sin 0° + cos²0° = 0² + 0 + 1² = 1, not 2.

Key idea to remember

Trigonometric identity: see Key ideas used on this page.

Explanation

An equation involving trigonometric ratios of an angle is a trigonometric identity when it is true for all values of the angle. Here sin²θ + cos²θ = 1 is such an identity. Using it turns the given expression into 1 + sin θ, and then sin θ = 1 gives θ = 90°.

Important formulas

sin²θ + cos²θ = 1 · Trigonometric identity that simplifies the given expression to 1 + sin θ.
sin 90° = 1 · Standard value used to identify θ = 90°.

Q26. If \(\sin^2 \theta = \frac{3}{4}\), then \(\theta\) is

Mark practised

(A) 30°
(B) 45°
(C) 60°
(D) 90°

Marks: 1
Level: Basic
Years: 2024
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2024 Mathematics Basic, Set 430/2/2, Q5, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 60°, because sin 60° = √3/2, so sin² 60° = (√3/2)² = 3/4, hence θ = 60°.
Why? Table 8.1 gives the exact sine values of the specific angles; squaring sin 60° reproduces the given value 3/4, so this angle satisfies the equation.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/2/2: option (C).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 30° because sin 30° = 1/2, without squaring it: sin² 30° = 1/4, not 3/4.
Taking the square root of 3/4 as 3/2 instead of √3/2.
Choosing 90° because sin 90° = 1, while sin² 90° = 1, not 3/4.

Why this works

All the options are acute angles, so sin θ is positive. Take the positive square root: sin θ = √(3/4) = √3/2. From Table 8.1, sin 60° = √3/2, so θ = 60°. Checking the other options: sin² 30° = (1/2)² = 1/4, sin² 45° = (1/√2)² = 1/2, sin² 90° = 1² = 1; none equals 3/4. Example 7 in the NCERT text uses the same idea: from sin R = 1/2 it finds angle PRQ = 30°.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

The NCERT text derives the trigonometric ratios of 30°, 45° and 60° using right triangles and lists them in Table 8.1. Here we match sin²θ = 3/4 with the square of the table value sin 60° = √3/2, so θ = 60°.

Important formulas

sin² θ = (sin θ)² · NCERT uses this notation for the square of sin θ.
sin 60° = √3/2 · From Table 8.1; squaring it gives 3/4.

Q27. If \(\cos \theta = \frac{1}{\sqrt{2}}\) then \(\tan \theta\) is equal to

Mark practised

(A) \(\frac{1}{\sqrt{2}}\)
(B) 0
(C) 1
(D) \(\sqrt{2} + 1\)

Marks: 1
Level: Basic
Years: 2024
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2024 Mathematics Basic, Set 430/3/3, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 1, because cos 45° = 1/√2 and tan 45° = 1.
Why? The table of specific angles gives cos 45° = 1/√2, so the angle is 45°, and the same table gives tan 45° = 1.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/3/3: option (C).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing option A because 1/√2 is the given value of cos θ, but the question asks for tan θ.
Choosing option B because 0 is the value of tan 0°, but the given cosine belongs to 45°, not 0°.
Choosing option D because √2 + 1 is formed from √2, but it is not a standard trigonometric ratio.

Why this works

For an acute angle, the equation cos θ = 1/√2 matches the table of specific angles only at θ = 45°. So tan θ = tan 45°. From the same table, tan 45° = 1. A quick check: sin 45° = 1/√2 and cos 45° = 1/√2, so tan 45° = (1/√2)/(1/√2) = 1. Option A is the cosine value itself, option B is tan 0°, and option D is not a standard ratio.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

This section gives the values of the six trigonometric ratios for 0°, 30°, 45°, 60° and 90° in Table 8.1. For 45°, the table gives cos 45° = 1/√2 and tan 45° = 1. The given cosine value matches only the 45° row, so θ = 45° and tan θ = 1.

Important formulas

cos 45° = 1/√2 · Value from Table 8.1 for the specific angle 45°.
tan 45° = 1 · Value from Table 8.1 for the specific angle 45°.
tan A = sin A / cos A · Used to check tan 45° = 1.

Q28. \(\left(\frac{1}{\cot^2 \theta}\right) - \left(\frac{1}{\cos^2 \theta}\right)\) is equal to :

Mark practised

(A) 1
(B) -1
(C) 0
(D) \(\sec^2 \theta\)

Marks: 1
Level: Basic
Years: 2024
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2024 Mathematics Basic, Set 430/4/2, Q12, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) −1, because 1/cot²θ − 1/cos²θ = tan²θ − sec²θ = −1.
Why? By the reciprocal definitions, 1/cot²θ = tan²θ and 1/cos²θ = sec²θ; the identity sec²θ − tan²θ = 1 then gives the value.

The full 1 mark is for the correct option.

Answer:

(B) −1

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/4/2: option (B).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Taking 1/cot²θ as cot²θ itself, so the difference becomes cot²θ − sec²θ.
Writing 1/cos²θ as cos²θ, so the difference becomes tan²θ − cos²θ.
Reversing the identity and thinking tan²θ − sec²θ equals 1, so choosing 1.

Why this works

Rewrite both fractions using reciprocal definitions. Since cot θ is the reciprocal of tan θ, 1/cot²θ = tan²θ. Since sec θ is the reciprocal of cos θ, 1/cos²θ = sec²θ. So the expression is tan²θ − sec²θ = −(sec²θ − tan²θ). The identity 1 + tan²θ = sec²θ gives sec²θ − tan²θ = 1, hence the value is −1. Quick check with θ = 45°: cot 45° = 1, so 1/cot²θ = 1, and cos² 45° = 1/2, so 1/cos²θ = 2; therefore 1 − 2 = −1.

Key idea to remember

Trigonometric identities: see Key ideas used on this page.

Explanation

A trigonometric identity is an equation involving trigonometric ratios of an angle that is true for all values of the angle for which the expressions are defined. This solution rewrites the fractions using reciprocal definitions and then applies the identity 1 + tan²θ = sec²θ.

Important formulas

tan θ = 1/cot θ · So 1/cot²θ = tan²θ.
sec θ = 1/cos θ · So 1/cos²θ = sec²θ.
sec²θ − tan²θ = 1 · Rearranged from the identity 1 + tan²θ = sec²θ.

Q29. For \(\theta = 30^\circ\), the value of \((2 \sin \theta \cos \theta)\) is :

Mark practised

(A) 1
(B) \(\frac{\sqrt{3}}{2}\)
(C) \(\frac{\sqrt{3}}{4}\)
(D) \(\frac{3}{2}\)

Marks: 1
Level: Standard
Years: 2024
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2024 Mathematics Standard, Set 30/1/3, Q9, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) √3/2, because 2 sin 30° cos 30° = 2 × 1/2 × √3/2 = √3/2.
Why? Substituting the fixed values of sin 30° and cos 30° into the expression gives its value.

The full 1 mark is for the correct option.

Answer:

(B) √3/2

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/1/3: option (B).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Taking sin 30° to be √3/2 and cos 30° to be 1/2, interchanging the two values.
Choosing √3/4 by forgetting to multiply by the 2 outside.

Q30. If \(x = a \cos \theta\) and \(y = b \sin \theta\) , then the value of \(b^2x^2 + a^2y^2\) is :

Mark practised

(A) \(a^2b^2\)
(B) \(ab\)
(C) \(a^4b^4\)
(D) \(a^2 + b^2\)

Marks: 1
Level: Standard
Years: 2024
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2024 Mathematics Standard, Set 30/5/2, Q5, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) a²b², because b²x² + a²y² = b²(a cos θ)² + a²(b sin θ)² = a²b²(cos²θ + sin²θ) = a²b².
Why? By the trigonometric identity cos²A + sin²A = 1, the bracket simplifies, so the whole expression reduces to a²b².

The full 1 mark is for the correct option.

Answer:

(A) a²b²

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/5/2: option (A).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Squaring only the ratio: writing b²x² as b²a cos θ instead of b²(a cos θ)² = a²b²cos²θ.
Choosing ab or a² + b² by matching letters, without substituting x = a cos θ and y = b sin θ first.
Replacing cos²θ + sin²θ by 0, or leaving the bracket unsimplified, instead of using the identity cos²θ + sin²θ = 1.

Why this works

Substitute first: b²x² = b²(a cos θ)² = a²b²cos²θ, and a²y² = a²(b sin θ)² = a²b²sin²θ. Adding gives a²b²(cos²θ + sin²θ). By the trigonometric identity cos²θ + sin²θ = 1, the bracket collapses, so the expression has the single value a²b². The other options are not obtained because the simplification leaves the product a²b², not ab, a⁴b⁴ or a² + b².

Key idea to remember

Trigonometric identity: see Key ideas used on this page.

Explanation

An equation involving trigonometric ratios of an angle is a trigonometric identity if it is true for all values of the angle. Here the identity cos²A + sin²A = 1 simplifies the bracket, so the whole expression becomes a²b².

Important formulas

cos²A + sin²A = 1 · Use it after substituting x = a cos θ and y = b sin θ.

Q31. If \(k \sin 30^\circ = \sqrt{2} \sin 45^\circ \tan 45^\circ\), then the value of 'k' is :

Mark practised

(A) 0
(B) 1
(C) \(\frac{1}{2}\)
(D) 2

Marks: 1
Level: Basic
Compartment: 2025
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2025 Compartment Mathematics Basic, Set 430/S/2, Q6, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 2, because k × 1/2 = √2 × 1/√2 × 1 = 1, so k = 2.
Why? Substitute the standard values from the table of trigonometric ratios of specific angles, then solve the linear equation for k.

The full 1 mark is for the correct option.

Answer:

(D) 2

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 430/S/2: option (D).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 1 because the right side equals 1, without dividing by sin 30°; the equation is k × (1/2) = 1, so k = 2.
Choosing 1/2 by treating k as the value of sin 30°; k is the unknown multiplier, not sin 30°.

Q32. The value of \(\frac{\sin^2 90^\circ + \cos^2 0^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}\) is :

Mark practised

(A) 1
(B) 0
(C) 2
(D) 4

Marks: 1
Level: Basic
Compartment: 2025
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2025 Compartment Mathematics Basic, Set 430/S/3, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 2, because sin² 90° + cos² 0° = 1 + 1 = 2 and sin² 30° + cos² 30° = 1/4 + 3/4 = 1.
Why? The standard values of trigonometric ratios of specific angles are substituted; squaring each value and simplifying gives the fraction.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 430/S/3: option (C).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Applying sin² A + cos² A = 1 to sin² 90° + cos² 0° and writing the numerator as 1, even though the identity needs the same angle in both terms.
Forgetting the squares and simplifying the denominator as 1/2 + √3/2 instead of 1/4 + 3/4.
Choosing 1 because the denominator is 1, without simplifying the numerator separately.

Why this works

Substitute the standard values of the specific angles. The numerator becomes 1 + 1 = 2, because sin 90° and cos 0° are both 1. The denominator becomes 1/4 + 3/4 = 1, after squaring the standard values of sin 30° and cos 30°. So the fraction is 2/1 = 2. Note that sin² A + cos² A = 1 holds only when both squares use the same angle A; here the numerator uses 90° and 0°, so the numerator is not 1.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

The section 'Trigonometric Ratios of Some Specific Angles' gives the exact values of sin and cos for 0°, 30°, 45°, 60° and 90°. The notation sin² A means (sin A)². Replace each ratio by its value, square it, and simplify the fraction.

Important formulas

sin² A = (sin A)² · The square applies to the value of the sine, not to the angle.
sin² A + cos² A = 1 · Trigonometric identity, valid when both terms use the same angle A.
sin² 90° = 1, cos² 0° = 1, sin² 30° = 1/4, cos² 30° = 3/4 · Standard values of trigonometric ratios of specific angles, after squaring.

Q33. If \(\cot A = \frac{7}{12}\) , then the value of \((\cos A + \sin A)\) cosec A is :

Mark practised

(A) \(\frac{5}{12}\)
(B) \(\frac{19}{12}\)
(C) \(\frac{19}{7}\)
(D) \(\frac{49}{144}\)

Marks: 1
Level: Standard
Compartment: 2025
Topic: 8.2 Trigonometric Ratios
Source: (CBSE 2025 Compartment Mathematics Standard, Set 30/S/2, Q14, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 19/12, because (cos A + sin A) cosec A = (cos A + sin A)/sin A = cot A + 1 = 7/12 + 1 = 19/12.
Why? cosec A is the reciprocal of sin A, and cot A = cos A/sin A, so the product simplifies to cot A + 1.

The full 1 mark is for the correct option.

Answer:

(B) 19/12

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 30/S/2: option (B).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Mistaking cosec A for sec A, so dividing by cos A instead of sin A; that leads to 1 + tan A = 1 + 12/7 = 19/7, option C.
Forgetting that sin A and cosec A cancel in the product, so giving 7/12 (cot A alone) instead of 19/12.

Why this works

The expression (cos A + sin A) cosec A is one product, but the definitions open it up. Since cosec A = 1/sin A, the product becomes (cos A + sin A) ÷ sin A. Dividing term by term gives cos A/sin A + 1, and cos A/sin A is cot A. So the expression is cot A + 1. With cot A = 7/12, this is 7/12 + 1 = 19/12. A quick check with a right triangle whose sides are in the ratio 7 : 12 (adjacent : opposite) gives hypotenuse √(7² + 12²), and (cos A + sin A) cosec A = (19/√(7² + 12²)) × (√(7² + 12²)/12) = 19/12.

Key idea to remember

Trigonometric ratios of an acute angle: see Key ideas used on this page.

Explanation

The trigonometric ratios of an acute angle relate the sides of a right triangle. The reciprocal relation cosec A = 1/sin A and the quotient relation cot A = cos A/sin A are enough to simplify the given expression without drawing the triangle.

Important formulas

cosec A = 1/sin A · Reciprocal definition of cosecant.
cot A = cos A/sin A · Quotient relation from the definitions.

Q34. The value of \((\tan A \csc A)^2 - (\sin A \sec A)^2\) is :

Mark practised

(A) 0
(B) 1
(C) -1
(D) 2

Marks: 1
Level: Standard
Years: 2025
Topic: 8.4 Trigonometric Identities
Source: (CBSE 2025 Mathematics Standard, Set 30/1/3, Q2, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 1, because tan A csc A = sec A and sin A sec A = tan A, so (tan A csc A)² − (sin A sec A)² = sec² A − tan² A = 1.
Why? tan A = sin A/cos A and csc A = 1/sin A make tan A csc A = sec A; also sin A sec A = tan A. The identity sec² A = 1 + tan² A gives sec² A − tan² A = 1.

The full 1 mark is for the correct option.

Answer:

(B) 1

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/1/3: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing csc A as 1/cos A; actually csc A is the reciprocal of sin A. With that wrong definition, tan A csc A = (sin A/cos A)(1/cos A) = sin A/cos² A, which at A = 45° equals √2, not 1.
Stopping at sec² A − tan² A and thinking the answer is 0 because sec A and tan A seem to cancel; the identity shows the value is 1.
Finding tan A csc A = sec A but forgetting to square it, and writing the answer as sec A − tan A instead of sec² A − tan² A.

Why this works

Write the ratios in terms of sin A and cos A. Since csc A = 1/sin A and sec A = 1/cos A, we get tan A csc A = (sin A/cos A) × (1/sin A) = 1/cos A = sec A, and sin A sec A = sin A × (1/cos A) = sin A/cos A = tan A. So the expression becomes sec² A − tan² A. From the identity sec² A = 1 + tan² A, it follows that sec² A − tan² A = 1. Quick check with a 3-4-5 triangle: tan A csc A = (3/4)(5/3) = 5/4 and sin A sec A = (3/5)(5/4) = 3/4, so (5/4)² − (3/4)² = 25/16 − 9/16 = 1.

Key idea to remember

Trigonometric Identities: see Key ideas used on this page.

Explanation

An equation involving trigonometric ratios of an angle is called a trigonometric identity if it is true for all values of the angle for which the expressions are defined. This solution uses the definitions tan A = sin A/cos A and csc A = 1/sin A, and the identity sec² A = 1 + tan² A, to simplify the given expression to 1.

Important formulas

tan A = sin A / cos A · Tangent expressed in terms of sine and cosine.
csc A = 1 / sin A, sec A = 1 / cos A · Reciprocal trigonometric ratios.
sec² A − tan² A = 1 · Rearranged form of the identity sec² A = 1 + tan² A.

Q35. If sin 4θ = √3/2, then θ/3 equals :

Mark practised

(A) 60°
(B) 20°
(C) 15°
(D) 5°

Marks: 1
Level: Standard
Years: 2025
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2025 Mathematics Standard, Set 30/3/2, Q5, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 5°, because sin 60° = √3/2, so 4θ = 60°, θ = 15° and θ/3 = 5°.
Why? From Table 8.1, the angle whose sine is √3/2 is the specific angle 60°, so the equation fixes 4θ; dividing by 4 and then by 3 gives θ/3.

The full 1 mark is for the correct option.

Answer:

(D) 5°

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/3/2: option (D).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 20° by dividing 60° by 3 directly, without first solving 4θ = 60° to get θ = 15°.
Choosing 15°, which is θ, instead of the asked value θ/3 = 5°.

Q36. The value of 2 tan 60° / 1 - tan² 60° is same as the value of

Mark practised

(A) -tan 30°
(B) -tan 60°
(C) 2 sin 60°
(D) 2 cos 60°

Marks: 1
Level: Standard
Years: 2025
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2025 Mathematics Standard, Set 30/5/2, Q12, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) -tan 60°, because 2 tan 60°/(1 - tan² 60°) = 2√3/(1 - 3) = -√3 = -tan 60°.
Why? A trigonometric ratio of a specific angle has a fixed value from the table; substituting it and simplifying the quotient gives the negative of that same ratio.

The full 1 mark is for the correct option.

Answer:

(B) -tan 60°

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/5/2: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Subtracting 3 from 1 as 2, thinking the denominator is positive, and then choosing 2 sin 60° because the value is taken as √3.
Reading the expression as (2 tan 60° ÷ 1) - tan² 60°, getting 2√3 - 3, and failing to match any option.
Choosing -tan 30° because it is negative, without checking that -tan 30° = -1/√3, which is not -√3.

Why this works

The fraction means (2 tan 60°) ÷ (1 - tan² 60°). From Table 8.1, tan 60° = √3. So the denominator is 1 - (√3)² = 1 - 3 = -2, and the fraction is 2√3 ÷ (-2) = -√3. Since tan 60° = √3, the value -√3 is exactly -tan 60°. Therefore option (B) is correct. The other options are different: -tan 30° = -1/√3, 2 sin 60° = √3, and 2 cos 60° = 1. A quick check: sin 60° = √3/2 and cos 60° = 1/2, so tan 60° = (√3/2) ÷ (1/2) = √3, matching Table 8.1.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

For the specific angles 0°, 30°, 45°, 60° and 90°, the trigonometric ratios have fixed values. Here tan 60° = √3 is substituted into the given fraction, and the simplification shows the fraction equals -tan 60°.

Important formulas

tan A = sin A / cos A · Definition used to verify tan 60° from sin 60° and cos 60°.
tan 60° = √3 · Value from Table 8.1, substituted in the question.

Q37. If \(x\left(\frac{2\tan 30^\circ}{1+\tan^2 30^\circ}\right) = y\left(\frac{2\tan 30^\circ}{1-\tan^2 30^\circ}\right)\), then \(x : y =\)

Mark practised

(A) 1 : 1
(B) 1 : 2
(C) 2 : 1
(D) 4 : 1

Marks: 1
Level: Standard
Years: 2025
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2025 Mathematics Standard, Set 30/6/1, Q12, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 2 : 1, because the first bracket is √3/2 and the second is √3, so x × √3/2 = y × √3 gives x : y = 2 : 1.
Why? Substituting the known value tan 30° = 1/√3 into both brackets gives their values, and the equation then becomes a simple proportion.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/6/1: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 1 : 2 by dividing x/2 = y wrongly and writing x = y/2 instead of x = 2y.
Cancelling 2 tan 30° from both sides without noticing that 1 + tan² 30° and 1 − tan² 30° are different denominators, so the two brackets are not equal.
Leaving the first bracket as 3/(2√3) without simplifying it to √3/2, which makes the ratio harder to see.

Why this works

Substitute tan 30° = 1/√3. First bracket: (2/√3) ÷ (1 + 1/3) = (2/√3) ÷ (4/3) = (2/√3) × (3/4) = 3/(2√3) = √3/2. Second bracket: (2/√3) ÷ (1 − 1/3) = (2/√3) ÷ (2/3) = √3. So x(√3/2) = y√3. Divide both sides by √3: x/2 = y, so x = 2y and x : y = 2 : 1. Quick check: with x = 2y, left side = 2y × √3/2 = y√3 and right side = y√3, so the equation balances.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

The angles 30°, 45° and 60° have fixed trigonometric values. Using tan 30° = 1/√3 lets us evaluate both brackets exactly, and the resulting equation directly gives the ratio x : y.

Important formulas

tan 30° = 1/√3 · Standard value used to evaluate both brackets.
2 tan 30° / (1 + tan² 30°) = √3/2 · This bracket equals sin 60°.
2 tan 30° / (1 − tan² 30°) = √3 · This bracket equals tan 60°.

Q38. If \(x = \cos 30^\circ - \sin 30^\circ\) and \(y = \tan 60^\circ - \cot 60^\circ\), then

Mark practised

(A) \(x = y\)
(B) \(x \gt y\)
(C) \(x \lt y\)
(D) \(x \gt 1, y \lt 1\)

Marks: 1
Level: Standard
Years: 2025
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2025 Mathematics Standard, Set 30/6/2, Q12, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) x < y, because x = cos 30° − sin 30° = (√3 − 1)/2 ≈ 0.366 and y = tan 60° − cot 60° = 2/√3 ≈ 1.155.
Why? Substituting the fixed standard values of the trigonometric ratios of 30° and 60° gives two numbers; the smaller one gives the relation.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/6/2: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Taking cot 60° as √3 instead of 1/√3, which makes y too small and can lead to choosing x > y.
Swapping the values of cos 30° and sin 30°, so that x becomes negative instead of positive.
Comparing only cos 30° with tan 60° and ignoring the subtracted terms; this is not a valid way to compare x and y.

Why this works

cos, sin, tan and cot are short forms of cosine, sine, tangent and cotangent; these are trigonometric ratios of an angle. The standard values are cos 30° = √3/2, sin 30° = 1/2, tan 60° = √3 and cot 60° = 1/√3. So x = √3/2 − 1/2 = (√3 − 1)/2 ≈ 0.366, and y = √3 − 1/√3 = 2/√3 ≈ 1.155. Since 0.366 < 1.155, x < y. Option D is false because x < 1 and y > 1.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

The values of the trigonometric ratios for 30°, 45° and 60° are fixed. This solution substitutes the values for 30° and 60° into the expressions for x and y, then compares the results.

Important formulas

cos 30° = √3/2 · Standard value for 30°.
sin 30° = 1/2 · Standard value for 30°.
tan 60° = √3 · Standard value for 60°.
cot 60° = 1/√3 · cot A is the reciprocal of tan A.
cot A = 1/tan A · Reciprocal relation between cotangent and tangent.

Q39. If \(x = 2 \sin 60^\circ \cos 60^\circ\) and \(y = \sin^2 30^\circ - \cos^2 30^\circ\) and \(x^2 = ky^2\), the value of \(k\) is

Mark practised

(A) \(\sqrt{3}\)
(B) \(-\sqrt{3}\)
(C) 3
(D) -3

Marks: 1
Level: Standard
Years: 2025
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2025 Mathematics Standard, Set 30/6/3, Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 3, because x = √3/2 and y = −1/2, so x² = 3/4 = 3y², hence k = 3.
Why? The standard trigonometric ratios of 30° and 60° are fixed values; substituting them into x and y and comparing their squares gives k.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/6/3: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing −3 because y is negative, forgetting that y² is positive.
Squaring √3/2 and writing 3/2 instead of 3/4.
Dividing x by y and getting −√3, instead of dividing x² by y².

Why this works

Use the specific-angle ratios: sin 60° is √3/2, cos 60° is 1/2, sin 30° is 1/2 and cos 30° is √3/2. Then x = 2 × (√3/2) × (1/2) = √3/2 and y = (1/2)² − (√3/2)² = 1/4 − 3/4 = −1/2. Squaring gives x² = 3/4 and y² = 1/4. The relation x² = k y² becomes 3/4 = k × (1/4), so k = 3. Since y² is positive, k must be positive, so the negative options are not possible. Another correct method also earns full marks.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

The ratios for 30°, 45° and 60° are fixed values from the specific-angles table. This question substitutes the 30° and 60° values into x and y, squares both, and uses x² = k y² to find k.

Important formulas

x = 2 sin 60° cos 60° · Given expression; substitute the standard values for 60°.
y = sin² 30° − cos² 30° · Given expression; substitute the standard values for 30°.
x² = k y² · Given relation; solve for k after squaring x and y.

Q40. If \(x \cdot \tan 45^\circ \cdot \sin 30^\circ = \cos 30^\circ \cdot \cot 60^\circ\), then \(x\) is equal to

Mark practised

(A) \(\sqrt{3}\)
(B) \(\frac{1}{\sqrt{3}}\)
(C) 1
(D) \(\frac{1}{2}\)

Marks: 1
Level: Standard
Years: 2026
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2026 Mathematics Standard, Set 30(B), Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 1, because x × 1 × 1/2 = (√3/2) × (1/√3) = 1/2, so x = 1.
Why? Each ratio in the equation has a fixed value from the table of specific angles; substituting those values changes the equation into a simple linear equation, whose solution is the value of x.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30(B): option (C).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Mistaking cot 60° for √3, which is tan 60°, gives an answer that is not in the options.
Forgetting to multiply by sin 30° gives 1/2 instead of 1.
Choosing √3 just because it appears on the right side, without simplifying (√3/2) × (1/√3).

Why this works

Use the standard values: tan 45° = 1, sin 30° = 1/2, cos 30° = √3/2 and cot 60° = 1/√3. Then the left side is x × 1 × 1/2 = x/2. The right side is (√3/2) × (1/√3) = 1/2. So x/2 = 1/2, which gives x = 1. Check: putting x = 1 makes both sides 1/2.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

The angles 30°, 45° and 60° have fixed trigonometric ratios, obtained from a right isosceles triangle and half an equilateral triangle. This solution replaces each ratio with its fixed value, so the trigonometric equation becomes an ordinary linear equation in x.

Important formulas

tan 45° = 1 · Standard value from the 45° right triangle.
sin 30° = 1/2 · Standard value from half an equilateral triangle.
cos 30° = √3/2 · Standard value from half an equilateral triangle.
cot 60° = 1/√3 · Reciprocal of tan 60° = √3.

Q41. The value of \(\left(\frac{1}{3}\cot^2 30^\circ - \frac{1}{2}\sec^2 60^\circ\right)\) is :

Mark practised

(A) -1
(B) -2
(C) \(\frac{5}{8}\)
(D) \(\frac{7}{8}\)

Marks: 1
Level: Standard
Years: 2026
Topic: 8.3 Trigonometric Ratios of Some Specific Angles
Source: (CBSE 2026 Mathematics Standard, Set 30/3/2, Q11, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) −1, because (1/3)(√3)² − (1/2)(2)² = 1 − 2 = −1.
Why? The standard values cot 30° = √3 and sec 60° = 2 are substituted, and cot² θ means (cot θ)², so each ratio is squared before multiplying by its coefficient.

The full 1 mark is for the correct option.

Answer:

(A) −1

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/3/2: option (A).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using the value of tan 30° for cot 30°.
Using the value of cos 60° for sec 60°.
Multiplying by the coefficient 1/3 before squaring cot 30°.

Why this works

The expression uses only two specific angles, 30° and 60°. The standard values are cot 30° = √3 and sec 60° = 2. Squaring first gives (√3)² = 3 and (2)² = 4. Then (1/3) × 3 = 1 and (1/2) × 4 = 2. So the expression becomes 1 − 2 = −1, which is option (A). A quick check: cot 30° is the reciprocal of tan 30°, and sec 60° is the reciprocal of cos 60°.

Key idea to remember

Trigonometric Ratios of Some Specific Angles: see Key ideas used on this page.

Explanation

This section gives the exact values of the trigonometric ratios for 30°, 45° and 60°. The solution substitutes cot 30° = √3 and sec 60° = 2, then squares and simplifies.

Important formulas

cot 30° = √3 · Standard value used in the first term.
sec 60° = 2 · Standard value used in the second term.
cot² θ = (cot θ)², sec² θ = (sec θ)² · The square applies to the whole ratio.

1 / 41

Key ideas used on this page

Trigonometric Ratios of Some Specific Angles
Used in questions 1 5 7 9 11 13 14 16 21 22 26 27 29 31 32 35 36 37 38 39 40 41
Trigonometric identity
Used in questions 2 8 12 18 19 20 23 25 30
Trigonometric Identities
Used in questions 4 28 34
Trigonometric ratios
Used in questions 6 17
Trigonometric identity: sin²θ + cos²θ = 1
Used in question 3
Trigonometric identity sec² A − tan² A = 1
Used in question 10
Trigonometric ratios of an angle
Used in question 15
Cosecant as the reciprocal of sine
Used in question 24
Trigonometric ratios of an acute angle
Used in question 33

How to use these questions

Try each question on paper first. Then read the steps: the mark beside each step shows what the examiner looks for. If a step is new to you, revise Section 8.2, Section 8.3 and Section 8.4 of the NCERT chapter.

To revise the lessons behind these questions, read the NCERT chapter, exactly as NCERT publishes it on its own website.

Official papers behind this page

Every question on this page comes from an official CBSE Class 10 Mathematics board paper. The board publishes the question papers on cbse.gov.in and the marking schemes on cbse.gov.in, where you can see how marks are given.

Year	Exam	Course	Question paper	Marking scheme
2026	Main	Standard (041)	Open CBSE's official question paper for the 2026 main exam (one ZIP file of about 61.9 MB, Class 10 Maths Standard and Basic together) to read the full paper.	Open CBSE's official marking scheme for the 2026 main exam (one ZIP file of about 22.8 MB, Class 10 Maths Standard (041)) to see how marks are given.
2025	Main	Standard (041)	Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 54.5 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given.
2025	Main	Basic (241)	Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 49.9 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given.
2025	Compartment	Standard (041)	Open CBSE's official question paper for the 2025 compartment exam (one ZIP file of about 3.4 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2025 compartment exam (one ZIP file of about 8.1 MB, Class 10 Maths Standard (041)) to see how marks are given.
2025	Compartment	Basic (241)	Open CBSE's official question paper for the 2025 compartment exam (one ZIP file of about 2.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2025 compartment exam (one ZIP file of about 18.7 MB, Class 10 Maths Basic (241)) to see how marks are given.
2024	Main	Standard (041)	Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 35.3 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.4 MB, Class 10 Maths Standard (041)) to see how marks are given.
2024	Main	Basic (241)	Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 43.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.8 MB, Class 10 Maths Basic (241)) to see how marks are given.
2024	Compartment	Basic (241)	Open CBSE's official question paper for the 2024 compartment exam (one ZIP file of about 2.2 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2024 compartment exam (one ZIP file of about 5.6 MB, Class 10 Maths Basic (241)) to see how marks are given.
2023	Main	Standard (041)	Open CBSE's official question paper for the 2023 main exam (one ZIP file of about 44 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2023 main exam (one ZIP file of about 17.4 MB, Class 10 Maths Standard (041)) to see how marks are given.
2023	Main	Basic (241)	Open CBSE's official question paper for the 2023 main exam (one ZIP file of about 49.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2023 main exam (one ZIP file of about 26 MB, Class 10 Maths Basic (241)) to see how marks are given.
2023	Compartment	Standard (041)	Open CBSE's official question paper for the 2023 compartment exam (one ZIP file of about 7.6 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2023 compartment exam (one ZIP file of about 3.3 MB, Class 10 Maths Standard (041)) to see how marks are given.

Sources and session

Each question is reproduced as printed in its CBSE paper, with its year, paper code and question number; only the layout is tidied. The same question printed in several paper sets counts once, and the card says how often it was printed in 2023–2026. These counts describe past papers; they are not predictions.

Every solution is original and follows NCERT Mathematics Class X (reprint 2026-27). The mark beside each step follows the official CBSE marking scheme where the scheme shows it; a mark labelled “(suggested)” follows the board’s usual pattern for that type of question.

Reference: CBSE Class 10 Mathematics board question papers via cbse.gov.in; all solutions are original. learncbse.net is an independent website and is not affiliated with CBSE or NCERT.

1-mark questions on Introduction to Trigonometry

All 41 Introduction to Trigonometry 1-mark questions

Q1. Which of the following statements is false ?

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q2. The value of \(\left( \cot^2 A - \frac{1}{\sin^2 A} \right)\) is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q3. \((\sin \theta + \cos \theta)^2 + (\sin \theta - \cos \theta)^2 =\)

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q4. \((\sec \theta - \cos \theta)^2 + \sin^2 \theta - \tan^2 \theta = ?\)

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q5. The value of \(\theta\) for which \(\sin 2\theta = \tan 45^\circ\) is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q6. If sin θ = cos θ, (0° < θ < 90°), then value of (sec θ · sin θ) is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q7. If \(\sin 30^\circ \tan 45^\circ = \frac{\sec 60^\circ}{k}\) , then the value of k is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q8. If \(\sec\theta - \tan\theta = 2\), then \(\sec\theta + \tan\theta\) is equal to :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q9. The value of \(2 \sin^2 30^\circ + 3 \tan^2 60^\circ - \cos^2 45^\circ\) is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q10. \(9 \sec^2 A - 9 \tan^2 A\) is equal to :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q11. [3/4 tan² 30° - sec² 45° + sin² 60°] is equal to

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q12. \(\frac{\cos^2 \theta}{\sin^2 \theta} - \frac{1}{\sin^2 \theta}\), in simplified form, is:

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q13. If \(\cos \theta = \frac{\sqrt{3}}{2}\) and \(\sin \phi = \frac{1}{2}\) , then \(\tan (\theta + \phi)\) is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q14. If \(\sin \theta = 1\), then the value of \(\frac{1}{2} \sin \left( \frac{\theta}{2} \right)\) is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q15. If \(4 \sec \theta - 5 = 0\) , then the value of \(\cot \theta\) is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q16. If \(\tan 3\theta = \sqrt{3}\), then \(\frac{\theta}{2}\) equals :

Step-by-step solution

Step 1: Choose the option