Prove the following identities (i) sec^4θ(1 - sec^4θ) - 2 tan^2θ = 1 (ii) (cotθ - cosθ )/(cotθ + cosθ) = (cosecθ - 1)/(cosecθ + 1)

Question

Prove the following identities

(i) sec⁴θ(1 - sec⁴θ) - 2 tan²θ = 1

(ii) \(\frac{cot\theta\,-\,cos\theta}{cot\theta\,+\,cos\theta}\) = \(\frac{cosec\theta\,-\,1}{cosec\theta\,+\,1}\)

Answer 1

(i) L.H.S. = sec⁴θ(1 - sec⁴θ) - 2 tan²θ = 1

(ii) \(\frac{cot\theta\,-\,cos\theta}{cot\theta\,+\,cos\theta}\)