Prove the following trigonometric identities.

We have to prove $\left(1+\tan ^{2} \theta\right)(1-\sin \theta)(1+\sin \theta)=1$

We know that,

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

$\sec ^{2} \theta-\tan ^{2} \theta=1$

So,

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

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

$=\sec ^{2} \theta \cos ^{2} \theta$

$=\frac{1}{\cos ^{2} \theta} \cos ^{2} \theta$

$=1$