Prove the following identities (1/(sec^2x - cos^2x) + 1/(cosec^2x - sin^2x)) sin^2x cos^2 x = (1 - sin^2x cos^2x)/(2 + sin^2x cos^2x)

Question

Prove the following identities

\(\Big(\cfrac1{sec^2\text x-cos^2\text x}+\cfrac1{cosec^2\text x-sin^2\text x}\Big)\)sin²x cos²x = \(\cfrac{1-sin^2\text x\,cos^2\text x}{2+sin^2\text x\,cos^2\text x}\)

Answer 1

LHS =  \(\Big(\cfrac1{sec^2\text x-cos^2\text x}+\cfrac1{cosec^2\text x-sin^2\text x}\Big)\)sin²x cos²x

We know that cosec θ = \(\cfrac1{sin\theta}\); sec θ = \(\cfrac1{cos\theta}\)

We know that sin²x + cos²x = 1.

= RHS

Hence proved.