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

Question
Express in terms of sin x: $$\displaystyle\frac{{{{\cot}^{{2}}{x}}-{1}}}{{{{\csc}^{{2}}{x}}}}$$

2020-12-02
$$\displaystyle\frac{{{{\cot}^{{2}}{\left({x}\right)}}-{1}}}{{{\csc}^{{2}}{\left({x}\right)}}}$$
$$\displaystyle=\frac{{{\left(\frac{{{\cos}^{{2}}{\left({x}\right)}}}{{{\sin}^{{2}}{\left({x}\right)}}}\right)}-{1}}}{{\frac{{1}}{{{\sin}^{{2}}{\left({x}\right)}}}}}$$
$$\displaystyle=\frac{{{\left({{\sin}^{{2}}{\left({x}\right)}}\right)}{\left({\left(\frac{{{\cos}^{{2}}{\left({x}\right)}}}{{{\sin}^{{2}}{\left({x}\right)}}}\right)}-{1}\right)}}}{{{\left({{\sin}^{{2}}{\left({x}\right)}}\right)}{\left(\frac{{1}}{{{\sin}^{{2}}{\left({x}\right)}}}\right)}}}$$
$$\displaystyle=\frac{{{{\cos}^{{2}}{\left({x}\right)}}-{{\sin}^{{2}}{\left({x}\right)}}}}{{1}}$$
$$\displaystyle={{\cos}^{{2}}{\left({x}\right)}}-{{\sin}^{{2}}{\left({x}\right)}}$$
$$\displaystyle={\left({1}-{{\sin}^{{2}}{\left({x}\right)}}\right)}-{{\sin}^{{2}}{\left({x}\right)}}$$
$$\displaystyle={1}-{2}{{\sin}^{{2}}{\left({x}\right)}}$$

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