Prove the following identities: cos x/(1 - sin x) = tan(π/4 + x/2)

Question

Prove the following identities:

\(\frac{cos\,x}{1-sin\,x} = tan \left(\frac{\pi}{4}+\frac{x}{2}\right)\)

Answer 1

To prove: \(\frac{cos\,x}{1-sin\,x} = tan \left(\frac{\pi}{4}+\frac{x}{2}\right)\)

Proof:

Take LHS:

\(\frac{cos\,x}{1-sin\,x}\)

Identities used:

Therefore,

Multiplying numerator and denominator by \(\frac{1}{\sqrt{2}}\):