Prove the following trigonometric identities.

We have to prove $(\operatorname{cosec} \theta+\sin \theta)(\operatorname{cosec} \theta-\sin \theta)=\cot ^{2} \theta+\cos ^{2} \theta$

We know that,

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

$\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$

So,

$(\operatorname{cosec} \theta+\sin \theta)(\operatorname{cosec} \theta-\sin \theta)=\operatorname{cosec}^{2} \theta-\sin ^{2} \theta$

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

$=1+\cot ^{2} \theta-1+\cos ^{2} \theta$

$=\cot ^{2} \theta+\cos ^{2} \theta$