Question

Prove the identity: 1 - (sin² x/(1 + cot x)) - (cos² x/(1 + tan x)) = sin x cos x 

Answer 1

Let us consider the LHS:

1 - (sin² x/(1 + cot x)) - (cos² x/(1 + tan x))

On using the formula,

tan θ = sin θ/cos θ;

cot θ = cos θ/sin θ

Then,

On using the formula, a³ + b³ = (a + b) (a² + b² – ab)

As we know, sin² x + cos² x = 1.

1 – 1 + sin x cos x

sin x cos x

= RHS

∴ LHS = RHS

Thus proved.