Exercise 8.2 (Trigonometry)

Evaluate the following :

(i) sin 60° cos 30° + sin 30° cos 60°

(ii) 2 tan² 45° + cos² 30° – sin² 60°

(iii) $\frac{\cos 45°}{\sec 30° + \mathrm{cosec} 30°}$

(iv) $\frac{\sin 30° + \tan 45° - \mathrm{cosec} 60°}{\sec 30° + \cos 60° + \cot 45°}$

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

Choose the correct option and justify your choice :

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

(ii) $\frac{1 - {\tan }^2 45°}{1 + {\tan }^2 45°}$ = 
(A) tan 90°
(B) 1
(C) sin 45°
(D) 0

(iii) sin 2A = 2 sin A is true when A =
(A) 0°
(B) 30°
(C) 45°
(D) 60°

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

If tan (A + B) = √3 and tan (A – B) = $\frac{1}{\sqrt{3}}$; 0° < A + B ≤ 90°; 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 θ increases as θ increases.
(iii) The value of cos θ increases as θ increases.
(iv) sin θ = cos θ for all values of θ.
(v) cot A is not defined for A = 0°.