Prove the following trigonometric identities.
Question:

Prove the following trigonometric identities.

$\cot ^{2} A \operatorname{cosec}^{2} B-\cot ^{2} B \operatorname{cosec}^{2} A=\cot ^{2} A-\cot ^{2} B$

Solution:

We have to prove $\cot ^{2} A \operatorname{cosec}^{2} B-\cot ^{2} B \operatorname{cosec}^{2} A=\cot ^{2} A-\cot ^{2} B$

We know that, $\operatorname{cosec}^{2} A-\cot ^{2} A=1$

So,

$\cot ^{2} A \operatorname{cosec}^{2} B-\cot ^{2} B \operatorname{cosec}^{2} A=\cot ^{2} A\left(1+\cot ^{2} B\right)-\cot ^{2} B\left(1+\cot ^{2} A\right)$

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

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

Hence proved.

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