Question:
Write the value of $\cot ^{2} \theta-\frac{1}{\sin ^{2} \theta}$.
Solution:
We have,
$\cot ^{2} \theta-\frac{1}{\sin ^{2} \theta}=\cot ^{2} \theta-\left(\frac{1}{\sin \theta}\right)^{2}$
$=\cot ^{2} \theta-(\operatorname{cosec} \theta)^{2}$
$=\cot ^{2} \theta-\operatorname{cosec}^{2} \theta$
We know that, $\cot ^{2} \theta-\operatorname{cosec}^{2} \theta=-1$
Therefore, $\cot ^{2} \theta-\frac{1}{\sin ^{2} \theta}=-1$
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