# How do you prove cot^2x - cos^2x = cot^2cos^2?

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#### Explanation:

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sjc Share
May 26, 2017

see below

#### Explanation:

to prove

${\cot}^{2} x - {\cos}^{2} x = {\cot}^{2} x {\cos}^{2} x$

take LHS and change to cosines an sines and then rearrange to arrive at the RHS

$= {\cos}^{2} \frac{x}{\sin} ^ 2 x - {\cos}^{2} x$

$= \frac{{\cos}^{2} x - {\cos}^{2} x {\sin}^{2} x}{\sin} ^ 2 x$

factorise numerator

$= \frac{{\cos}^{2} x \left(1 - {\sin}^{2} x\right)}{\sin} ^ 2 x$

$\implies \frac{{\cos}^{2} x \cdot {\cos}^{2} x}{\sin} ^ 2 x$

$= {\cos}^{2} x \cdot \left({\cos}^{2} \frac{x}{\sin} ^ 2 x\right)$

$= {\cos}^{2} x {\cot}^{2} x = {\cot}^{2} x {\cos}^{2} x$

=RHS as reqd.

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