Prove the following trigonometric identities.

We have to prove $(\sec \theta+\cos \theta)(\sec \theta-\cos \theta)=\tan ^{2} \theta+\sin ^{2} \theta$

We know that,

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

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

$(\sec \theta+\cos \theta)(\sec \theta-\cos \theta)=\sec ^{2} \theta-\cos ^{2} \theta$

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

$=1+\tan ^{2} \theta-1+\sin ^{2} \theta$

$=\tan ^{2} \theta+\sin ^{2} \theta$