NCERT Solutions
Class 10 - Mathematics - Chapter 8: INTRODUCTION TO TRIGONOMETRY

Exercise 8.2

Evaluate the following:

1. \(\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ\)
2. \(2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ\)
3. \(\dfrac{\cos 45^\circ}{\sec 30^\circ + \text{cosec } 30^\circ}\)
4. \(\dfrac{\sin 30^\circ + \tan 45^\circ - \text{cosec } 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}\)
5. \(\dfrac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}\)

Choose the correct option and justify your choice:

1. \(\dfrac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} =\)
   (A) \(\sin 60^\circ\)   (B) \(\cos 60^\circ\)   (C) \(\tan 60^\circ\)   (D) \(\sin 30^\circ\)
2. \(\dfrac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ} =\)
   (A) \(\tan 90^\circ\)   (B) 1   (C) \(\sin 45^\circ\)   (D) 0
3. \(\sin 2A = 2 \sin A\) is true when \(A =\)
   (A) \(0^\circ\)   (B) \(30^\circ\)   (C) \(45^\circ\)   (D) \(60^\circ\)
4. \(\dfrac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} =\)
   (A) \(\cos 60^\circ\)   (B) \(\sin 60^\circ\)   (C) \(\tan 60^\circ\)   (D) \(\sin 30^\circ\)

If \(\tan (A + B) = \sqrt{3}\) and \(\tan (A - B) = \dfrac{1}{\sqrt{3}}\); \(0^\circ < A + B \leq 90^\circ\), \(A > B\), find \(A\) and \(B\).

State whether the following are true or false. Justify your answer.

1. \(\sin (A + B) = \sin A + \sin B\).
2. The value of \(\sin \theta\) increases as \(\theta\) increases.
3. The value of \(\cos \theta\) increases as \(\theta\) increases.
4. \(\sin \theta = \cos \theta\) for all values of \(\theta\).
5. \(\cot A\) is not defined for \(A = 0^\circ\).