Question

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)

Answer 1

`\bf \textit{Cofunction Identities} \\ \quad \\ sin\left((\pi)/(2)-{{ \theta}}\right)=cos({{ \theta}})\qquad \boxed{cos\left((\pi)/(2)-{{ \theta}}\right)=sin({{ \theta}})} \\ \quad \\ \quad \\ tan\left((\pi)/(2)-{{ \theta}}\right)=cot({{ \theta}})\qquad cot\left((\pi)/(2)-{{ \theta}}\right)=tan({{ \theta}}) \\ \quad \\ \quad \\ sec\left((\pi)/(2)-{{ \theta}}\right)=csc({{ \theta}})\qquad csc\left((\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 )=√(1-sin^2(\theta ))}`


`\bf \\\\ -------------------------------\\\\ \cfrac{cos^2\left((\pi )/(2)-x \right)}{√(1-sin^2(x))}\implies \cfrac{\left[ cos\left((\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)`