Prove the following trigonometric identities.

In the given question, we need to prove $\frac{1+\cos \theta-\sin ^{2} \theta}{\sin \theta(1+\cos \theta)}=\cot \theta$.

Using the property $\sin ^{2} \theta+\cos ^{2} \theta=1$, we get

So,

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

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

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

Solving further, we get

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

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

$=\cot \theta$

Hence proved.