# How to use trigonometric identities to simplify 1 ( 1 &#x2212;<

How to use trigonometric identities to simplify $\frac{1}{\left(1-\mathrm{csc}x\right)\left(1+\mathrm{csc}x\right)}$?
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Ariella Bruce
$\frac{1}{\left(1-\mathrm{csc}x\right)\left(1+\mathrm{csc}x\right)}=\frac{1}{1-{\mathrm{csc}}^{2}x}=\frac{1}{1-\frac{1}{{\mathrm{sin}}^{2}x}}=\frac{{\mathrm{sin}}^{2}x}{{\mathrm{sin}}^{2}x-1}=-\frac{{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}=-{\mathrm{tan}}^{2}x$