Question

If A + B + C = 270°, then cos 2A + cos 2B + cos 2C is equal to ______.

Options

• 4 sin A sin B sin C

• 4 cos A cos B cos C

• 1 – 4 sin A sin B sin C

• 1 – 4 cos A cos B cos C

Answer 1

If A + B + C = 270°, then cos 2A + cos 2B + cos 2C is equal to 1 – 4 sin A sin B sin C.

Explanation:

cos 2A + cos 2B + cos 2C

= 2 cos (A + B) cos (A – B) + 1 – 2 sin² C

= `2 cos ((3π)/2 - C) cos (A - B) + 1 - 2 sin^2 C`  ...`[∵ A + B + C = 270^circ \implies B + A = (3π)/2 - C]`

= 1 – 2 sin C [cos (A – B) + sin C]

= 1 – 2 sin C [cos (A – B) – cos (A + B)]

= 1 – 4 sin A sin B sin C