Question

If A + B = C, then cos² A + cos² B + cos² C – 2 cos A cos B cos C is equal to ______.

Options

• 1

• 2

• 0

• 3

Answer 1

If A + B = C, then cos² A + cos² B + cos² C – 2 cos A cos B cos C is equal to 1.

Explanation:

cos² A + cos² B + cos² C

= `(1 + cos 2A)/2 + (1 + cos 2B)/2 + cos^2 C`

= `1 + 1/2 (cos 2A + cos 2B) + cos^2 C`

= `1 + 2/2 [cos(A + B) cos(A - B) + cos^2 C`

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

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

= 1 + 2 cos C cos B cos A

`\implies` cos² A + cos² B + cos² C – 2 cos A cos B cos C = 1