# Can you prove? Cosec^4(1+cos^2x)(1-cos^2x)-2cot^2x=1

## Can you prove? $C o {\sec}^{4} x \left(1 + {\cos}^{2} x\right) \left(1 - {\cos}^{2} x\right) - 2 {\cot}^{2} x = 1$.

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Feb 2, 2018

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$\textcolor{red}{{\csc}^{4} x} \left(1 + {\cos}^{2} x\right) \textcolor{red}{\left(1 - {\cos}^{2} x\right)} - 2 {\cot}^{2} x$,

$= \textcolor{red}{{\csc}^{4} x} \left(1 + {\cos}^{2} x\right) \textcolor{red}{{\sin}^{2} x} - 2 {\cot}^{2} x$,

$= \textcolor{red}{{\csc}^{2} x} \left(1 + {\cos}^{2} x\right) - 2 {\cot}^{2} x$,

$= \textcolor{red}{{\csc}^{2} x} + {\csc}^{2} x {\cos}^{2} x - 2 {\cot}^{2} x$,

$= \textcolor{red}{1 + {\cot}^{2} x} + \textcolor{g r e e n}{\frac{1}{\sin} ^ 2 x \cdot {\cos}^{2} x} - 2 {\cot}^{2} x$,

$= 1 + \cancel{{\cot}^{2} x + \textcolor{g r e e n}{{\cot}^{2} x} - 2 {\cot}^{2} x}$,

$= 1$.

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