Prove the following trigonometric identities.

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

Multiplying numerator and denominator under the square root by $(1-\cos \theta)$, we have

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

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

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

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

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

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