Question

The value of 2 sin 2 B + 4 cos (A + B) sin A sin B + cos 2 (A + B) is

A. 0

B. cos 3 A

C. cos 2A

D. None of these

Answer 1

Correct answer is C.

Given expression is

2 sin² B + 4 cos (A + B) sin A sin B + cos 2 (A + B)

[using the cos (A+B) = cos A cos B – sin A sin B]

= 2 sin² B + 4 sin A sin B [cos A cos B - sin A sin B] + cos 2 (A + B)

= 2 sin² B + 4 sin A sin B cos A cos B - 4 sin A sin B sin A sin B + cos 2 (A + B)

= 2 sin² B + (2 sin A cos A) (2sin B cos B) - 4 sin² A sin² B + cos 2 (A + B)

[ using sin 2A = 2 sin A cos A]

Hence

2 sin² B + 4 cos (A + B) sin A sin B + cos 2 (A + B) = cos 2A