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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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