# Express in terms of sin x: (cot^2x-1)/(csc^2x)

Express in terms of sin x: $\frac{{\mathrm{cot}}^{2}x-1}{{\mathrm{csc}}^{2}x}$
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bahaistag
$\frac{{\mathrm{cot}}^{2}\left(x\right)-1}{{\mathrm{csc}}^{2}\left(x\right)}$
$=\frac{\left(\frac{{\mathrm{cos}}^{2}\left(x\right)}{{\mathrm{sin}}^{2}\left(x\right)}\right)-1}{\frac{1}{{\mathrm{sin}}^{2}\left(x\right)}}$
$=\frac{\left({\mathrm{sin}}^{2}\left(x\right)\right)\left(\left(\frac{{\mathrm{cos}}^{2}\left(x\right)}{{\mathrm{sin}}^{2}\left(x\right)}\right)-1\right)}{\left({\mathrm{sin}}^{2}\left(x\right)\right)\left(\frac{1}{{\mathrm{sin}}^{2}\left(x\right)}\right)}$
$=\frac{{\mathrm{cos}}^{2}\left(x\right)-{\mathrm{sin}}^{2}\left(x\right)}{1}$
$={\mathrm{cos}}^{2}\left(x\right)-{\mathrm{sin}}^{2}\left(x\right)$
$=\left(1-{\mathrm{sin}}^{2}\left(x\right)\right)-{\mathrm{sin}}^{2}\left(x\right)$
$=1-2{\mathrm{sin}}^{2}\left(x\right)$