Prove the following trigonometric identities.
Question:

Prove the following trigonometric identities.

$\sin ^{2} A \cos ^{2} B-\cos ^{2} A \sin ^{2} B=\sin ^{2} A-\sin ^{2} B$

Solution:

We know that, $\sin ^{2} A+\cos ^{2} A=1$

So have,

$\sin ^{2} A \cos ^{2} B-\cos ^{2} A \sin ^{2} B=\sin ^{2} A\left(1-\sin ^{2} B\right)-\left(1-\sin ^{2} A\right) \sin ^{2} B$

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

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

Hence proved.

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