Prove the following identities (cos x)/(1- sin x) = (1 + cos x + sin x)/(1 + cos x - sin x)

Question

Prove the following identities

\(\cfrac{cos\,\text x}{1-sin\,\text x}\) = \(\cfrac{1+cos\,\text x+sin\,\text x}{1+cos\,\text x-sin\,\text x}\)

Answer 1

RHS = \(\cfrac{1+cos\,\text x+sin\,\text x}{1+cos\,\text x-sin\,\text x}\) 

We know that sin²x + cos²x = 1.

We know that 1 – cos²x = sin²x.

We know that 1 – sin²x = cos²x.

= LHS

Hence proved.