Question

Prove the following identities:

`sec^2θ - (sin^2θ - 2sin^4θ)/(2cos^4θ - cos^2θ) = 1`

Answer 1

LHS = `(1 + tan^2θ) - (sin^2θ(1 - 2sin^2θ))/(cos^2θ(2cos^2θ - 1))`

= (1 + tan²θ) – tan²θ   ...[∵ 1 – 2 sin²θ = 2 cos²θ – 1 = 1 – cos 2θ]

= 1 = RHS.