Simplify: $\frac{cos^{2}\theta}{cosec^{2}\theta.sin^{2}\theta-\sin^{2}\theta}$.

Given: $\frac{cos^{2}\theta}{cosec^{2}\theta.sin^{2}\theta-\sin^{2}\theta}$.

To do: To simplify: $\frac{cos^{2}\theta}{cosec^{2}\theta.sin^{2}\theta-\sin^{2}\theta}$.
 

Solution:

$\frac{cos^{2}\theta}{cosec^{2}\theta.sin^{2}\theta-\sin^{2}\theta}$

$=\frac{cos^{2}\theta}{sin^{2}\theta( cosec^{2}\theta-1)}$

$=\frac{cos^{2}\theta}{sin^{2}\theta}.\frac{1}{( cosec^{2}\theta-1)}$

$=cot^2\theta.\frac{1}{cot^2\theta}$       [$\because \frac{cos^{2}\theta}{sin^{2}\theta}=cot^2\theta$ and $cosec^2\theta-1=cot^2\theta$ ]

$=1$

Thus, $\frac{cos^{2}\theta}{cosec^{2}\theta.sin^{2}\theta-\sin^{2}\theta}=1$.

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Updated on: 10-Oct-2022

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