Question

Prove that `(1 + 1/(tan^2θ))(1 + 1/(cos^2θ)) = 1/(sin^2θ - sin^4θ)`.

Answer 1

L.H.S. = `(1 + 1/(tan^2θ))(1 + 1/(cot^2θ))`

= `((1 + cot^2θ)(1 + tan^2θ))/(cot^2θ . tan^2θ)`   ...[∵ cosec² θ – cot² θ = 1, sec² θ – tan² θ = 1]

= cosec² θ · sec² θ   ...(cot² θ . tan² θ = 1)

= `1/(sin^2θ) * 1/(cos^2θ)`   ...(∵ cos² θ = 1 – sin² θ)

= `1/(sin^2θ(1 - sin^2θ))`

= `1/(sin^2θ - sin^4θ)`

L.H.S. = R.H.S.

Hence proved.