Prove the following trigonometric identities.

Question:
Prove the following trigonometric identities.

$\tan ^{2} A \sec ^{2} B-\sec ^{2} A \tan ^{2} B=\tan ^{2} A-\tan ^{2} B$

Solution:

We have to prove $\tan ^{2} A \sec ^{2} B-\sec ^{2} A \tan ^{2} B=\tan ^{2} A-\tan ^{2} B$

We know that, $\sec ^{2} A-\tan ^{2} A=1$

So,

$\tan ^{2} A \sec ^{2} B-\sec ^{2} A \tan ^{2} B=\tan ^{2} A\left(1+\tan ^{2} B\right)-\left(1+\tan ^{2} A\right) \tan ^{2} B$

$=\tan ^{2} A+\tan ^{2} A \tan ^{2} B-\tan ^{2} B-\tan ^{2} A \tan ^{2} B$

$=\tan ^{2} A-\tan ^{2} B$

Hence proved.