Question

Prove that:

tan(π + x) cot(x – π) – cos(2π – x) cos(2π + x) = sin² x.

Answer 1

tan(π + x) cot(x – π) – cos(2π – x) cos(2π + x) = (tan x) (-cot(π – x) – cos x cos x

[∵ cot(x – π) = cot(-(π – x)) = -cot(π – x) = cot x]

= tan x cot x – cos² x

= 1 – cos² x

= sin² x [∵ sin² x + cos² x = 1 ⇒ sin² x = (1 – cos² x)]