demilon
28

# simplify the expression. cos^2(pi/2-x)/sqrt1-sin^2(x) a. tan(x) b. cos(x)tan(x) c. cos(x)cot(x) d. sin(x)tan(x)

$\bf \textit{Cofunction Identities} \\ \quad \\ sin\left(\frac{\pi}{2}-{{ \theta}}\right)=cos({{ \theta}})\qquad \boxed{cos\left(\frac{\pi}{2}-{{ \theta}}\right)=sin({{ \theta}})} \\ \quad \\ \quad \\ tan\left(\frac{\pi}{2}-{{ \theta}}\right)=cot({{ \theta}})\qquad cot\left(\frac{\pi}{2}-{{ \theta}}\right)=tan({{ \theta}}) \\ \quad \\ \quad \\ sec\left(\frac{\pi}{2}-{{ \theta}}\right)=csc({{ \theta}})\qquad csc\left(\frac{\pi}{2}-{{ \theta}}\right)=sec({{ \theta}})$ $\bf \\\\ -------------------------------\\\\ sin^2(\theta)+cos^2(\theta)=1\implies cos^2(\theta)=1-sin^2(\theta ) \\\\\\ \boxed{cos(\theta )=\sqrt{1-sin^2(\theta )}}$ $\bf \\\\ -------------------------------\\\\ \cfrac{cos^2\left(\frac{\pi }{2}-x \right)}{\sqrt{1-sin^2(x)}}\implies \cfrac{\left[ cos\left(\frac{\pi }{2}-x \right)\right]^2}{cos(x)}\implies \cfrac{[sin(x)]^2}{cos(x)}\implies \cfrac{sin(x)sin(x)}{cos(x)} \\\\\\ sin(x)\cdot \cfrac{sin(x)}{cos(x)}\implies sin(x)tan(x)$