# Proving the identity csc &#x2061;<!-- ⁡ --> x &#x2212; sin &#x2061;<!-- ⁡ -->

Proving the identity $\mathrm{csc}x-\mathrm{sin}x=\left(\mathrm{cot}x\right)\left(\mathrm{cos}x\right)$
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So starting with LHS:
$\begin{array}{rl}\frac{1}{\mathrm{sin}x}-\mathrm{sin}x& =\frac{1-\left(\mathrm{sin}x{\right)}^{2}}{\mathrm{sin}x}\\ & =\frac{\left(\mathrm{cos}x{\right)}^{2}}{\mathrm{sin}x}\end{array}$
$\frac{\mathrm{cos}x.\mathrm{cos}x}{\mathrm{sin}x}$
$=\frac{\mathrm{cos}x}{\mathrm{sin}x}.\mathrm{cos}x$
$=\mathrm{cot}x.\mathrm{cos}x$