Question

Simplify the expression: cos^2(pi-x)/sqrt1-sin^2(x) A. Tan(x) B. Cos(x) tan(x) C. Cos(x) cot(x) D. Sin(x) tan(x)

Answer 1

$\bf sin^2(\theta)+cos^2(\theta)=1\implies cos^2(\theta)=1-sin^2(\theta) \\\\\\ {cos(\theta )=\sqrt{1-sin^2(\theta)}}\qquad\qquad and \\\\\\ {cos({{ \alpha}} - {{ \beta}})= cos({{ \alpha}})cos({{ \beta}}) + sin({{ \alpha}})sin({{ \beta}})}\\\\ -------------------------------$

$\bf \cfrac{cos^2(\pi -x)}{\sqrt{1-sin^2(x)}}\implies \cfrac{[cos(\pi -x)]^2}{cos(x)} \\\\\\ \cfrac{[cos(\pi )cos(x)-sin(\pi )sin(x)]^2}{cos(x)}\qquad \begin{cases} cos(\pi )=-1\\ sin(\pi )=0 \end{cases}\qquad thus \\\\\\ \cfrac{[\boxed{-1}cos(x)-\boxed{0}sin(x)]^2}{cos(x)}\implies \cfrac{[-cos(x)]^2}{cos(x)}\implies \cfrac{[cos(x)]^2}{cos(x)} \\\\\\ \cfrac{cos^2(x)}{cos(x)}\implies cos(x)$