Prove the following identities (i) (1 - tan^2θ)/(cot^2θ - 1) = tan^2θ (ii) cosθ/(1 + sinθ) = secθ - tanθ

Question

Prove the following identities

(i) \(\frac{1\,-\,tan^2\theta}{cot^2\theta\,-\,1}\) = tan²θ

(ii) \(\frac{cos\theta}{1\,+\,sin\theta}\) = secθ - tanθ 

Answer 1

(i) L.H.S. = \(\frac{1\,-\,tan^2\theta}{cot^2\theta\,-\,1}\)

(ii) L.H.S = \(\frac{cos\theta}{1\,+\,sin\theta}\) 

(Multiplying Numerator and denominator by (1 - sinθ))