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.