Exercise 8.2

Evaluate the following :

(i) $\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ$

(ii) $2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ$

(iii) $\dfrac{\cos 45^\circ}{\sec 30^\circ} + \csc 30^\circ$

(iv) $\dfrac{\sin 30^\circ + \tan 45^\circ - \csc 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}$

(v) $\dfrac{2 \sin 60^\circ + 5 \cos 60^\circ}{4 \sec 30^\circ - \tan 45^\circ} + \dfrac{\sin 30^\circ}{\cos 30^\circ}$

Choose the correct option and justify your choice :

(i) $\dfrac{2\tan 30^\circ}{1+\tan^2 30^\circ} =$ (A) $\sin 60^\circ$    (B) $\cos 60^\circ$    (C) $\tan 60^\circ$    (D) $\sin 30^\circ$

(ii) $\dfrac{1-\tan 45^\circ}{1+\tan 45^\circ} =$ (A) $\tan 90^\circ$    (B) $1$    (C) $\sin 45^\circ$    (D) $0$

(iii) $\sin 2A = 2 \sin A$ is true when $A =$ (A) $0^\circ$    (B) $30^\circ$    (C) $45^\circ$    (D) $60^\circ$

(iv) $\dfrac{2\tan 30^\circ}{1-\tan^2 30^\circ} =$ (A) $\cos 60^\circ$    (B) $\sin 60^\circ$    (C) $\tan 60^\circ$    (D) $\sin 30^\circ$

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

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

(i) $\sin(A + B) = \sin A + \sin B$.

(ii) The value of $\sin \theta$ increases as $\theta$ increases.

(iii) The value of $\cos \theta$ increases as $\theta$ increases.

(iv) $\sin \theta = \cos \theta$ for all values of $\theta$.

(v) $\cot A$ is not defined for $A = 0^\circ$.