Prove the following trigonometric identities.

We have to prove $\frac{\sin \theta}{1-\cos \theta}=\operatorname{cosec} \theta+\cot \theta$.

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

Multiplying both numerator and denominator by $(1+\cos \theta)$, we have

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

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

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

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

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

$=\operatorname{cosec} \theta+\cot \theta$