# Rewrite the expression in terms of the given function. 1/(1+cos x)+cos x/(1-cos x);cot x

Kymani Hatfield 2022-10-22 Answered
Rewrite the expression in terms of the given function.
$\frac{1}{1+\mathrm{cos}x}+\frac{\mathrm{cos}x}{1-\mathrm{cos}x};\mathrm{cot}x$
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honejata1
$\frac{1}{1+\mathrm{cos}x}+\frac{\mathrm{cos}x}{1-\mathrm{cos}x};\mathrm{cot}x\phantom{\rule{0ex}{0ex}}=\frac{1-\mathrm{cos}x+\left(1+\mathrm{cos}x\right)\left(\mathrm{cos}x\right)}{1-{\mathrm{cos}}^{2}x}\phantom{\rule{0ex}{0ex}}=\frac{1+{\mathrm{cos}}^{2}x}{1-{\mathrm{cos}}^{2}x}\phantom{\rule{0ex}{0ex}}=\frac{1+{\mathrm{cos}}^{2}x}{{\mathrm{sin}}^{2}x}\phantom{\rule{0ex}{0ex}}=\frac{1}{{\mathrm{sin}}^{2}x}+\frac{{\mathrm{cos}}^{2}x}{{\mathrm{sin}}^{2}x}\phantom{\rule{0ex}{0ex}}={\mathrm{csc}}^{2}x+{\mathrm{cot}}^{2}x\phantom{\rule{0ex}{0ex}}=\left(1+{\mathrm{cot}}^{2}x\right)+\mathrm{cot}x\phantom{\rule{0ex}{0ex}}=1+2{\mathrm{cot}}^{2}x$