Question

csc(-x)/1+tan^2x) = ?

Answer 1

$\bf 1+tan^2(\theta)=sec^2(\theta)\qquad \qquad sin(-\theta )=-sin(\theta ) \\\\\\ cot(\theta)=\cfrac{cos(\theta)}{sin(\theta)}\qquad \qquad csc(\theta)=\cfrac{1}{sin(\theta)} \qquad \qquad % secant sec(\theta)=\cfrac{1}{cos(\theta)}\\\\ -------------------------------$

$\bf \cfrac{csc(-x)}{1+tan^2(x)} \implies \cfrac{csc(-x)}{sec^2(x)}\implies \cfrac{\frac{1}{sin(-x)}}{\frac{1^2}{cos^2(x)}} \\\\\\ \cfrac{1}{-sin(x)}\cdot \cfrac{cos^2(x)}{1}\implies -\cfrac{cos^2(x)}{sin(x)}\implies cos(x)\cfrac{cos(x)}{sin(x)} \\\\\\ \boxed{cos(x)cot(x)}$