Mathematics · Textbook solutions

Introduction to Trigonometry

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 59 questions

8.2 Trigonometric Ratios

23 q

Solved Examples

Worked · 5

8.1 Eg.1 $\tan \text{A} = \dfrac{4}{3}$
8.1 Eg.2 $\angle$
8.1 Eg.3 $\triangle$
8.1 Eg.4 $\tan \text{A} = 1$
8.1 Eg.5 $\triangle$

Exercise 8.1

Practice · 18

In $\triangle$ ABC, right-angled at B, AB $= 24$ cm, BC $= 7$ cm. Determine :
Ex 8.1 Q1 (i) $\sin \text{A}$
Ex 8.1 Q1 (ii) $\sin \text{C}$
Ex 8.1 Q2 $\tan \text{P} - \cot \text{R}$
Ex 8.1 Q3 $\sin \text{A} = \dfrac{3}{4}$
Ex 8.1 Q4 $15 \cot \text{A} = 8$
Ex 8.1 Q5 $\sec\theta = \dfrac{13}{12}$
Ex 8.1 Q6 $\angle$
Ex 8.1 Q8 $3 \cot \text{A} = 4$
Ex 8.1 Q10 $\triangle$
If $\cot\theta = \dfrac{7}{8}$ , evaluate :
Ex 8.1 Q7 (i) $\dfrac{(1 + \sin\theta)(1 - \sin\theta)}{(1 + \cos\theta)(1 - \cos\theta)}$
Ex 8.1 Q7 (ii) $\cot^2\theta$
In triangle ABC, right-angled at B, if $\tan \text{A} = \dfrac{1}{\sqrt{3}}$ , find the value of:
Ex 8.1 Q9 (i) $\sin \text{A} \cos \text{C} + \cos \text{A} \sin \text{C}$
Ex 8.1 Q9 (ii) $\cos \text{A} \cos \text{C} - \sin \text{A} \sin \text{C}$
State whether the following are true or false. Justify your answer.
Ex 8.1 Q11 (i) $\tan \text{A}$
Ex 8.1 Q11 (ii) $\sec \text{A} = \dfrac{12}{5}$
Ex 8.1 Q11 (iii) $\cos \text{A}$
Ex 8.1 Q11 (iv) $\cot \text{A}$
Ex 8.1 Q11 (v) $\sin\theta = \dfrac{4}{3}$

8.3 Trigonometric Ratios of Some Specific Angles

18 q

Solved Examples

Worked · 3

8.2 Eg.6 $\triangle$
8.2 Eg.7 $\triangle$
8.2 Eg.8 $\sin(\text{A} - \text{B}) = \dfrac{1}{2}$

Exercise 8.2

Practice · 15

Evaluate the following :
Ex 8.2 Q1 (i) $\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ$
Ex 8.2 Q1 (ii) $2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ$
Ex 8.2 Q1 (iii) $\dfrac{\cos 45^\circ}{\sec 30^\circ + \operatorname{cosec} 30^\circ}$
Ex 8.2 Q1 (iv) $\dfrac{\sin 30^\circ + \tan 45^\circ - \operatorname{cosec} 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}$
Ex 8.2 Q1 (v) $\dfrac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$
Choose the correct option and justify your choice :
Ex 8.2 Q2 (i) $\dfrac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} =$
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Ex 8.2 Q2 (ii) $\dfrac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ} =$
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Ex 8.2 Q2 (iii) $\sin 2\text{A} = 2 \sin \text{A}$
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Ex 8.2 Q2 (iv) $\dfrac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} =$
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Ex 8.2 Q3 $\tan(\text{A} + \text{B}) = \sqrt{3}$
State whether the following are true or false. Justify your answer.
Ex 8.2 Q4 (i) $\sin(\text{A} + \text{B}) = \sin \text{A} + \sin \text{B}$
Ex 8.2 Q4 (ii) $\sin\theta$
Ex 8.2 Q4 (iii) $\cos\theta$
Ex 8.2 Q4 (iv) $\sin\theta = \cos\theta$
Ex 8.2 Q4 (v) $\cot \text{A}$

8.4 Trigonometric Identities

18 q

Solved Examples

Worked · 4

8.3 Eg.9 $\cos \text{A}$
8.3 Eg.10 $\sec \text{A}\,(1 - \sin \text{A})(\sec \text{A} + \tan \text{A}) = 1$
8.3 Eg.11 $\dfrac{\cot \text{A} - \cos \text{A}}{\cot \text{A} + \cos \text{A}} = \dfrac{\operatorname{cosec} \text{A} - 1}{\operatorname{cosec} \text{A} + 1}$
8.3 Eg.12 $\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \dfrac{1}{\sec\theta - \tan\theta}$

Exercise 8.3

Practice · 14

Ex 8.3 Q1 $\sin \text{A}$
Ex 8.3 Q2 $\angle$
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Ex 8.3 Q3 (i) $9 \sec^2 \text{A} - 9 \tan^2 \text{A} =$
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Ex 8.3 Q3 (ii) $(1 + \tan\theta + \sec\theta)(1 + \cot\theta - \operatorname{cosec}\theta) =$
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Ex 8.3 Q3 (iii) $(\sec \text{A} + \tan \text{A})(1 - \sin \text{A}) =$
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Ex 8.3 Q3 (iv) $\dfrac{1 + \tan^2 \text{A}}{1 + \cot^2 \text{A}} =$
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Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
Ex 8.3 Q4 (i) $(\operatorname{cosec}\theta - \cot\theta)^2 = \dfrac{1 - \cos\theta}{1 + \cos\theta}$
Ex 8.3 Q4 (ii) $\dfrac{\cos \text{A}}{1 + \sin \text{A}} + \dfrac{1 + \sin \text{A}}{\cos \text{A}} = 2 \sec \text{A}$
Ex 8.3 Q4 (iii) $\dfrac{\tan\theta}{1 - \cot\theta} + \dfrac{\cot\theta}{1 - \tan\theta} = 1 + \sec\theta \operatorname{cosec}\theta$
Ex 8.3 Q4 (iv) $\dfrac{1 + \sec \text{A}}{\sec \text{A}} = \dfrac{\sin^2 \text{A}}{1 - \cos \text{A}}$
Ex 8.3 Q4 (v) $\dfrac{\cos \text{A} - \sin \text{A} + 1}{\cos \text{A} + \sin \text{A} - 1} = \operatorname{cosec} \text{A} + \cot \text{A}$
Ex 8.3 Q4 (vi) $\sqrt{\dfrac{1 + \sin \text{A}}{1 - \sin \text{A}}} = \sec \text{A} + \tan \text{A}$
Ex 8.3 Q4 (vii) $\dfrac{\sin\theta - 2\sin^3\theta}{2\cos^3\theta - \cos\theta} = \tan\theta$
Ex 8.3 Q4 (viii) $(\sin \text{A} + \operatorname{cosec} \text{A})^2 + (\cos \text{A} + \sec \text{A})^2 = 7 + \tan^2 \text{A} + \cot^2 \text{A}$

What the board has asked from this chapter

Questions per paper · 63 papers · 2023 to 2026

2026

44 q

3 $\cot\theta = 3$
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Trigonometric Ratios of an Acute Angle
6 $\theta$
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Trigonometric Ratios of an Acute Angle
9 $\dfrac{\sec^2 A - 1}{\sin^2 A}$
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Trigonometric Identities
9 $\sin\theta = \dfrac{a}{b}$
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Trigonometric Ratios of an Acute Angle
10 $\cos A = \dfrac{1}{2}$
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Trigonometric Ratios of Specific Angles
10 $\cos y = 0$
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Trigonometric Ratios of Specific Angles
10 $\cos A = \dfrac{4}{5}$
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Trigonometric Ratios of an Acute Angle
11 $\left(\dfrac{1}{3}\cot^2 30^\circ - \dfrac{1}{2}\sec^2 60^\circ\right)$
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Trigonometric Ratios of Specific Angles
11 $2\sin A = 1$
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Trigonometric Ratios of Specific Angles
11 $\sin A = \dfrac{1}{3}$
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Trigonometric Ratios of an Acute Angle
12 $\dfrac{1 + \tan^2 A}{1 + \cot^2 A}$
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Trigonometric Identities
14 $\theta$
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Trigonometric Ratios of an Acute Angle
15 $2\tan A = 3$
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Trigonometric Ratios of an Acute Angle
15 $\sin 2\alpha = \dfrac{\sqrt{3}}{2}$
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Trigonometric Ratios of Specific Angles
18 $\dfrac{\sec A}{\sqrt{\sec^2 A - 1}}$
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Trigonometric Identities
18 $\left(\dfrac{1}{2}\tan^2 45^\circ - \cos^2 60^\circ\right)$
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Trigonometric Ratios of Specific Angles
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled as Assertion (A) and the other is labelled as Reason (R). Select the correct answer to these questions from the codes (A), (B), (C) and (D) as given below.
20 $\tan 2\theta$
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Trigonometric Ratios of Specific Angles
21 (a) $\dfrac{\tan\theta}{1 + \tan^2\theta} + \dfrac{\cot\theta}{1 + \cot^2\theta} = 2\sin\theta\cos\theta$
Trigonometric Identities
21 (b) $\dfrac{1 - 2\tan^2 30^\circ - \sec^2 45^\circ}{\sin^2 60^\circ}$
Trigonometric Ratios of Specific Angles
22 (a) $\sec(2A - B) = \sqrt{2}$
Trigonometric Ratios of Specific Angles
22 (b) $\dfrac{2\cos 30^\circ - \cot^3 60^\circ}{\tan 30^\circ}$
Trigonometric Ratios of Specific Angles
22 (a) $\dfrac{5\cos^2 60^\circ + 4\sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$
Trigonometric Ratios of Specific Angles
23 (a) $\sqrt{\dfrac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A$
Trigonometric Identities
22 (b) $1 + \dfrac{\cot^2\alpha}{1 + \operatorname{cosec}\alpha} = \operatorname{cosec}\alpha$
Trigonometric Identities
23 (b) $\dfrac{3\cos^2 30^\circ - 6\operatorname{cosec}^2 30^\circ}{\tan^2 60^\circ}$
Trigonometric Ratios of Specific Angles
24 (a) $\tan\theta + \dfrac{1}{\tan\theta} = 2$
Trigonometric Identities
24 (a) $\dfrac{\sin^3 60^\circ - \tan 30^\circ}{\cos^2 45^\circ}$
Trigonometric Ratios of Specific Angles
24 (A) $\tan\theta = \dfrac{24}{7}$
Trigonometric Ratios of an Acute Angle
25 (a) $\tan A = \dfrac{4}{3}$
Trigonometric Ratios of an Acute Angle
24 (b) $A + 2B$
Trigonometric Ratios of Specific Angles
24 (b) $\sqrt{\dfrac{1 - \sin\theta}{1 + \sin\theta}} = \sec\theta - \tan\theta$
Trigonometric Identities
24 (B) $\cot\theta = \dfrac{7}{8}$
Trigonometric Identities
25 (b) $\cos A$
Trigonometric Identities
26 (a) $\sin\theta + \cos\theta = \sqrt{3}$
Trigonometric Identities
26 (b) $(\sin A + \sec A)^2 + (\cos A + \operatorname{cosec} A)^2 = (1 + \sec A \operatorname{cosec} A)^2$
Trigonometric Identities
28 (a) $\dfrac{\sec^3\theta}{\sec^2\theta - 1} + \dfrac{\operatorname{cosec}^3\theta}{\operatorname{cosec}^2\theta - 1} = \sec\theta \cdot \operatorname{cosec}\theta\,(\sec\theta + \operatorname{cosec}\theta)$
Trigonometric Identities
28 (A) $x = h + a\cos\theta$
Trigonometric Identities
28 (B) $\dfrac{\tan A}{1 + \sec A} - \dfrac{\tan A}{1 - \sec A} = 2\operatorname{cosec} A$
Trigonometric Identities
28 $\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \dfrac{1}{\sec\theta - \tan\theta}$
Trigonometric Identities
28 (b) $\dfrac{\sec\alpha}{\operatorname{cosec}\beta} = p$
Trigonometric Identities
28 $\dfrac{\tan\theta}{1 - \cot\theta} + \dfrac{\cot\theta}{1 - \tan\theta} = 1 + \tan\theta + \cot\theta$
Trigonometric Identities
30 $\dfrac{1}{\sec x - \tan x} - \dfrac{1}{\cos x} = \dfrac{1}{\cos x} - \dfrac{1}{\sec x + \tan x}$
Trigonometric Identities
31 $\sec\theta + \tan\theta = m$
Trigonometric Identities
31 $(1 + \cot\theta - \operatorname{cosec}\theta)(1 + \tan\theta + \sec\theta) = 2$
Trigonometric Identities

2025

53 q

1 $\tan 3\theta = \sqrt{3}$
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Trigonometric Ratios of Specific Angles
1 $7\cos^2\theta + 3\sin^2\theta = 4$
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Trigonometric Identities
2 $\sin\theta = \cos\theta\ (0^\circ < \theta < 90^\circ)$
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Trigonometric Ratios of Specific Angles
2 $(\tan A \operatorname{cosec} A)^2 - (\sin A \sec A)^2$
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Trigonometric Identities
3 $x = 2 \sin 60^\circ \cos 60^\circ$
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Trigonometric Ratios of Specific Angles
3 $(1 - 2\sin^2 60^\circ)$
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Trigonometric Ratios of Specific Angles
4 $\theta$
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Trigonometric Ratios of Specific Angles
5 $\sin 4\theta = \dfrac{\sqrt{3}}{2}$
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Trigonometric Ratios of Specific Angles
5 $\tan^2\theta - \left(\dfrac{1}{\cos\theta} \times \sec\theta\right)$
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Trigonometric Identities
7 $\alpha + \beta = 90^\circ$
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Trigonometric Ratios of Specific Angles
8 $\tan 2A = 3\tan A$
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Trigonometric Ratios of Specific Angles
9 $(\cot\theta + \tan\theta)$
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Trigonometric Identities
9 $Which of the following statements is true ?$
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Trigonometric Ratios of an Acute Angle
10 $Which of the following is a trigonometric identity ?$
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Trigonometric Identities
11 $\sin (\alpha + \beta) = 1$
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Trigonometric Ratios of Specific Angles
12 $\dfrac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ}$
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Trigonometric Ratios of Specific Angles
12 $x = \cos 30^\circ - \sin 30^\circ$
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Trigonometric Ratios of Specific Angles
12 $x\left(\dfrac{2\tan 30^\circ}{1 + \tan^2 30^\circ}\right) = y\left(\dfrac{2\tan 30^\circ}{1 - \tan^2 30^\circ}\right)$
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Trigonometric Ratios of Specific Angles
12 $\dfrac{2 \tan 60^\circ}{1 - \tan^2 60^\circ}$
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Trigonometric Ratios of Specific Angles
13 $\sin 30^\circ \tan 45^\circ = \dfrac{\sec 60^\circ}{k}$
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Trigonometric Ratios of Specific Angles
13 $\dfrac{\cos\theta}{\sqrt{1 - \cos^2\theta}}$
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Trigonometric Identities
17 $\sec A = 2\cos A$
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Trigonometric Ratios of Specific Angles
18 $\sin B = \dfrac{1}{4}$
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Trigonometric Ratios of an Acute Angle
Directions : In Question Numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option from following :
19 $\theta$
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Trigonometric Identities
Directions : In Question Numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option from following :
19 $\theta$
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Trigonometric Ratios of an Acute Angle
Directions : In Question Numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option from following :
20 $\theta$
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Trigonometric Identities
21 $4k = \tan^2 60^\circ - 2\operatorname{cosec}^2 30^\circ - 2\tan^2 30^\circ$
Trigonometric Ratios of Specific Angles
21 (a) $x\cos 60^\circ + y\cos 0^\circ + \sin 30^\circ - \cot 45^\circ = 5$
Trigonometric Ratios of Specific Angles
21 $\tan A + \cot A = 6$
Trigonometric Identities
21 (b) $\dfrac{\tan^2 60^\circ}{\sin^2 60^\circ + \cos^2 30^\circ}$
Trigonometric Ratios of Specific Angles
22 (a) $a\sec\theta + b\tan\theta = m$
Trigonometric Identities
22 (b) $\sin^2 A + \cos^2 A = 1$
Trigonometric Identities
23 (a) $\sin(A - B) = \sin A \cos B - \cos A \sin B$
Trigonometric Ratios of Specific Angles
23 $\tan A = \sqrt{3}$
Trigonometric Ratios of Specific Angles
23 (b) $\sin A = y$
Trigonometric Identities
24 (a) $(\sin A + \operatorname{cosec} A)^2 + (\cos A + \sec A)^2 = x + \tan^2 A + \cot^2 A$
Trigonometric Identities
24 (b) $\dfrac{3\sin 30^\circ - 4\sin^3 30^\circ}{2\sin^2 50^\circ + 2\cos^2 50^\circ}$
Trigonometric Ratios of Specific Angles
26 (a) $\sqrt{\sec^2\theta + \operatorname{cosec}^2\theta} = \tan\theta + \cot\theta$
Trigonometric Identities
26 (a) $\tan\theta + \sin\theta = m$
Trigonometric Identities
26 (b) $\dfrac{\cot A - 1}{2 - \sec^2 A} = \dfrac{\cot A}{1 + \tan A}$
Trigonometric Identities
26 (b) $\operatorname{cosec}\theta = x + \dfrac{1}{4x}$
Trigonometric Identities
27 (a) $\dfrac{\tan\theta}{1 - \cot\theta} + \dfrac{\cot\theta}{1 - \tan\theta} = 1 + \sec\theta\operatorname{cosec}\theta$
Trigonometric Identities
27 (a) $\left(1 + \dfrac{1}{\tan^2\theta}\right)\left(1 + \dfrac{1}{\cot^2\theta}\right) = \dfrac{1}{\sin^2\theta - \sin^4\theta}$
Trigonometric Identities
27 (b) $\dfrac{\sin A + \cos A}{\sin A - \cos A} + \dfrac{\sin A - \cos A}{\sin A + \cos A} = \dfrac{2}{2\sin^2 A - 1}$
Trigonometric Identities
27 (b) $\sqrt{\dfrac{\operatorname{cosec}\theta - 1}{\operatorname{cosec}\theta + 1}} + \sqrt{\dfrac{\operatorname{cosec}\theta + 1}{\operatorname{cosec}\theta - 1}} = 2\sec\theta$
Trigonometric Identities
28 (a) $\sqrt{\dfrac{\sec A - 1}{\sec A + 1}} + \sqrt{\dfrac{\sec A + 1}{\sec A - 1}} = 2\operatorname{cosec} A$
Trigonometric Identities
28 (b) $\left(\dfrac{1}{\cos A} - \cos A\right)\left(\dfrac{1}{\sin A} - \sin A\right) = \dfrac{1}{\tan A + \cot A}$
Trigonometric Identities
29 (a) $\dfrac{\cos A + \sin A - 1}{\cos A - \sin A + 1} = \operatorname{cosec} A - \cot A$
Trigonometric Identities
30 (a) $\dfrac{1 + \operatorname{cosec} A}{\operatorname{cosec} A} = \dfrac{\cos^2 A}{1 - \sin A}$
Trigonometric Identities
29 (b) $\cot\theta + \cos\theta = p$
Trigonometric Identities
30 (a) $\dfrac{\cos\theta - 2\cos^3\theta}{\sin\theta - 2\sin^3\theta} + \cot\theta = 0$
Trigonometric Identities
30 (b) $\sin\theta + \cos\theta = x$
Trigonometric Identities
30 (b) $2A + B$
Trigonometric Ratios of Specific Angles

2024

37 q

2 $\dfrac{x}{3} = 2\sin A$
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Trigonometric Identities
3 $5\tan\theta - 12 = 0$
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Trigonometric Ratios of an Acute Angle
3 $4\sec\theta - 5 = 0$
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Trigonometric Ratios of an Acute Angle
4 $\sin A = \dfrac{2}{3}$
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Trigonometric Ratios of an Acute Angle
5 $x = a\cos\theta$
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Trigonometric Identities
7 $\sec\theta - \tan\theta = m$
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Trigonometric Identities
7 $\sin\theta = \cos\theta$
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Trigonometric Ratios of Specific Angles
8 $\cos\theta = \dfrac{\sqrt{3}}{2}$
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Trigonometric Ratios of Specific Angles
9 $\theta = 30^\circ$
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Trigonometric Ratios of Specific Angles
10 $\sin\theta = 1$
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Trigonometric Ratios of Specific Angles
12 $\cos(\alpha + \beta) = 0$
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Trigonometric Ratios of Specific Angles
13 $\sin\alpha = \dfrac{\sqrt{3}}{2}$
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Trigonometric Ratios of Specific Angles
18 $\theta$
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Trigonometric Ratios of Specific Angles
Directions : In Question 19 and 20, Assertion (A) and Reason (R) are given. Select the correct option from the following :
19 $\sin A = \dfrac{1}{3}$
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Trigonometric Identities
21 $\dfrac{\sec^2 45^\circ - \tan^2 45^\circ}{\sin^2 45^\circ}$
Trigonometric Ratios of Specific Angles
23 (A) $2\sqrt{2}\cos 45^\circ \sin 30^\circ + 2\sqrt{3}\cos 30^\circ$
Trigonometric Ratios of Specific Angles
22 $\dfrac{5\tan 60^\circ}{(\sin^2 60^\circ + \cos^2 60^\circ)\tan 30^\circ}$
Trigonometric Ratios of Specific Angles
23 (a) $2\sin^2 30^\circ \sec 60^\circ + \tan^2 60^\circ$
Trigonometric Ratios of Specific Angles
22 $\dfrac{2\tan 30^\circ \cdot \sec 60^\circ \cdot \tan 45^\circ}{1 - \sin^2 60^\circ}$
Trigonometric Ratios of Specific Angles
23 (b) $2\sin(A + B) = \sqrt{3}$
Trigonometric Ratios of Specific Angles
23 (B) $A = 60^\circ$
Trigonometric Ratios of Specific Angles
23 (a) $\dfrac{5\cos^2 60^\circ + 4\sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \sin^2 60^\circ}$
Trigonometric Ratios of Specific Angles
23 (b) $\sin(A - B) = \dfrac{1}{2}$
Trigonometric Ratios of Specific Angles
25 $\dfrac{\cos 45^\circ + \sin 60^\circ}{\sec 30^\circ + \operatorname{cosec} 30^\circ}$
Trigonometric Ratios of Specific Angles
26 $\dfrac{\tan\theta - \cot\theta}{\sin\theta \cos\theta} = \sec^2\theta - \operatorname{cosec}^2\theta$
Trigonometric Identities
26 $\dfrac{\operatorname{cosec}^2\theta - \sec^2\theta}{\operatorname{cosec}^2\theta + \sec^2\theta} = \dfrac{3}{4}$
Trigonometric Ratios of an Acute Angle
26 $\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \dfrac{1}{\sec\theta - \tan\theta}$
Trigonometric Identities
27 $\dfrac{\sin A + \cos A}{\sin A - \cos A} + \dfrac{\sin A - \cos A}{\sin A + \cos A} = \dfrac{2}{2\sin^2 A - 1}$
Trigonometric Identities
28 $\dfrac{\tan\theta}{1 - \cot\theta} + \dfrac{\cot\theta}{1 - \tan\theta} = 1 + \sec\theta \operatorname{cosec}\theta$
Trigonometric Identities
29 $\sin\theta + \cos\theta = p$
Trigonometric Identities
29 $\dfrac{1 + \sec\theta - \tan\theta}{1 + \sec\theta + \tan\theta} = \dfrac{1 - \sin\theta}{\cos\theta}$
Trigonometric Identities
29 $\dfrac{\tan A}{1 - \cot A} + \dfrac{\cot A}{1 - \tan A} = 1 + \sec A \operatorname{cosec} A$
Trigonometric Identities
30 $\dfrac{\sin\theta - 2\sin^3\theta}{2\cos^3\theta - \cos\theta} = \tan\theta$
Trigonometric Identities
30 $\dfrac{\sin\theta}{1 + \cos\theta} + \dfrac{1 + \cos\theta}{\sin\theta} = 2\operatorname{cosec}\theta$
Trigonometric Identities
31 $\sqrt{\sec^2\theta + \operatorname{cosec}^2\theta} = \tan\theta + \cot\theta$
Trigonometric Identities
31 $\sin^6\theta + \cos^6\theta = 1 - 3\sin^2\theta\cos^2\theta$
Trigonometric Identities
31 $(\operatorname{cosec}\theta - \sin\theta)(\sec\theta - \cos\theta)(\tan\theta + \cot\theta) = 1$
Trigonometric Identities

2023

37 q

1 $\left(\dfrac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ}\right)$
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Trigonometric Ratios of Specific Angles
1 $\theta$
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Trigonometric Identities
2 $\tan\theta = \dfrac{x}{y}$
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Trigonometric Ratios of an Acute Angle
6 $\tan\theta = \dfrac{5}{12}$
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Trigonometric Ratios of an Acute Angle
6 $2\tan A = 3$
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Trigonometric Ratios of an Acute Angle
7 $\dfrac{\cos^2\theta}{\sin^2\theta} - \dfrac{1}{\sin^2\theta}$
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Trigonometric Identities
8 $(\sec^2\theta - 1)(\operatorname{cosec}^2\theta - 1)$
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Trigonometric Identities
9 $\sec\theta - \tan\theta = \dfrac{1}{3}$
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Trigonometric Identities
9 $\left[\dfrac{3}{4}\tan^2 30^\circ - \sec^2 45^\circ + \sin^2 60^\circ\right]$
Tap an option to check your answer.
Trigonometric Ratios of Specific Angles
10 $\sec\theta$
Tap an option to check your answer.
Trigonometric Identities
12 $\left(\dfrac{2\tan 30^\circ}{1 + \tan^2 30^\circ}\right)$
Tap an option to check your answer.
Trigonometric Ratios of Specific Angles
16 $\theta$
Tap an option to check your answer.
Trigonometric Identities
17 $\left[\dfrac{5}{8}\sec^2 60^\circ - \tan^2 60^\circ + \cos^2 45^\circ\right]$
Tap an option to check your answer.
Trigonometric Ratios of Specific Angles
18 $(\cos^4 A - \sin^4 A)$
Tap an option to check your answer.
Trigonometric Identities
In question numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option out of the following :
20 $0 < \theta \leq 90^\circ$
Tap an option to check your answer.
Trigonometric Identities
21 (a) $\sin\theta + \sin^2\theta = 1$
Trigonometric Identities
21 (A) $\dfrac{5}{\cot^2 30^\circ} + \dfrac{1}{\sin^2 60^\circ} - \cot^2 45^\circ + 2\sin^2 90^\circ$
Trigonometric Ratios of Specific Angles
21 (B) $\theta$
Trigonometric Ratios of Specific Angles
22 (a) $4\cot^2 45^\circ - \sec^2 60^\circ + \sin^2 60^\circ + p = \dfrac{3}{4}$
Trigonometric Ratios of Specific Angles
21 (b) $\tan\theta = \dfrac{1}{\sqrt{7}}$
Trigonometric Identities
22 (b) $\cos A + \cos^2 A = 1$
Trigonometric Identities
22 (A) $2\sec^2\theta + 3\operatorname{cosec}^2\theta - 2\sin\theta\cos\theta$
Trigonometric Ratios of Specific Angles
23 (a) $\sin\theta + \cos\theta = \sqrt{3}$
Trigonometric Identities
22 (B) $\sin\theta - \cos\theta = 0$
Trigonometric Ratios of Specific Angles
23 (b) $\sin\alpha = \dfrac{1}{\sqrt{2}}$
Trigonometric Ratios of Specific Angles
25 (A) $\dfrac{5\cos^2 60^\circ + 4\sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$
Trigonometric Ratios of Specific Angles
25 (a) $a\cos\theta + b\sin\theta = m$
Trigonometric Identities
25 (B) $\sin(A - B) = 0$
Trigonometric Ratios of Specific Angles
26 $(\sin\theta + \cos\theta)(\tan\theta + \cot\theta) = \sec\theta + \operatorname{cosec}\theta$
Trigonometric Identities
26 $\dfrac{\cos^2\theta}{1 - \tan\theta} + \dfrac{\sin^3\theta}{\sin\theta - \cos\theta} = 1 + \sin\theta\cos\theta$
Trigonometric Identities
27 $2(\sin^6\theta + \cos^6\theta) - 3(\sin^4\theta + \cos^4\theta) + 1 = 0$
Trigonometric Identities
27 $\left(\dfrac{1}{\cos\theta} - \cos\theta\right)\left(\dfrac{1}{\sin\theta} - \sin\theta\right) = \dfrac{1}{\tan\theta + \cot\theta}$
Trigonometric Identities
29 (A) $\dfrac{\sin A - 2\sin^3 A}{2\cos^3 A - \cos A} = \tan A$
Trigonometric Identities
29 (B) $\sec A(1 - \sin A)(\sec A + \tan A) = 1$
Trigonometric Identities
30 $\dfrac{\tan\theta + \sec\theta - 1}{\tan\theta - \sec\theta + 1} = \dfrac{1 + \sin\theta}{\cos\theta}$
Trigonometric Identities
30 $\dfrac{1 + \sec A}{\sec A} = \dfrac{\sin^2 A}{1 - \cos A}$
Trigonometric Identities
31 $(\operatorname{cosec} A - \sin A)(\sec A - \cos A) = \dfrac{1}{\cot A + \tan A}$
Trigonometric Identities

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 4 times: 2023*, 2024, 2025*, 2026 (* = numbers or wording changed that year)
28 $\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \dfrac{1}{\sec\theta - \tan\theta}$
Question 2
Asked 3 times: 2024, 2025, 2026* (* = numbers or wording changed that year)
28 $\dfrac{\tan\theta}{1 - \cot\theta} + \dfrac{\cot\theta}{1 - \tan\theta} = 1 + \tan\theta + \cot\theta$
Question 3
Asked 3 times: 2023*, 2024*, 2026 (* = numbers or wording changed that year)
22 (a) $\sec(2A - B) = \sqrt{2}$
Question 4
Asked 3 times: 2024*, 2025*, 2026 (* = numbers or wording changed that year)
10 $\cos y = 0$
Tap an option to check your answer.
Question 5
Asked 3 times: 2023, 2024*, 2026 (* = numbers or wording changed that year)
22 (a) $\dfrac{5\cos^2 60^\circ + 4\sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$
Question 6
Asked 3 times: 2023, 2024*, 2025 (* = numbers or wording changed that year)
28 (b) $\left(\dfrac{1}{\cos A} - \cos A\right)\left(\dfrac{1}{\sin A} - \sin A\right) = \dfrac{1}{\tan A + \cot A}$
Question 7
Asked 3 times: 2023, 2024, 2025* (* = numbers or wording changed that year)
30 (a) $\dfrac{\cos\theta - 2\cos^3\theta}{\sin\theta - 2\sin^3\theta} + \cot\theta = 0$
Question 8
Asked 2 times: 2023*, 2026 (* = numbers or wording changed that year)
26 (a) $\sin\theta + \cos\theta = \sqrt{3}$
Question 9
Asked 2 times: 2025*, 2026 (* = numbers or wording changed that year)
25 (b) $\cos A$
Question 10
Asked 2 times: 2024*, 2026 (* = numbers or wording changed that year)
11 $\sin A = \dfrac{1}{3}$
Tap an option to check your answer.

8.2 Trigonometric Ratios

Solved Examples

Exercise 8.1

8.3 Trigonometric Ratios of Some Specific Angles

Solved Examples

Exercise 8.2

8.4 Trigonometric Identities

Solved Examples

Exercise 8.3

What the board has asked from this chapter

2026

2025

2024

2023

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10