Prove the following trigonometric identities.

We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$

Multiplying the both numerator and the denominator by $(1-\sin \theta)$, we have

$\frac{\cos \theta}{1+\sin \theta}=\frac{\cos \theta(1-\sin \theta)}{(1+\sin \theta)(1-\sin \theta)}$

$=\frac{\cos \theta(1-\sin \theta)}{\left(1-\sin ^{2} \theta\right)}$

$=\frac{\cos \theta(1-\sin \theta)}{\cos ^{2} \theta}$

$=\frac{1-\sin \theta}{\cos \theta}$