Write the value of cot2 θ−1sin2 θ.

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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