Question

Prove the identity csc^2xsec^2x=sec^2x+csc^2x

Answer 1

`\bf csc(\theta)=\cfrac{1}{sin(\theta)} \qquad % secant sec(\theta)=\cfrac{1}{cos(\theta)} \\\\\\ and\qquad sin^2(\theta)+cos^2(\theta)=1 \\\\ -------------------------------\\\\ csc^2(x)sec^2(x)=sec^2(x)+csc^2(x)\\\\ -------------------------------\\\\ sec^2(x)+csc^2(x)\implies \cfrac{1}{cos^2(x)}+\cfrac{1}{sin^2(x)}\implies \cfrac{sin^2(x)+cos^2(x)}{cos^2(x)sin^2(x)} \\\\\\ \cfrac{1}{cos^2(x)sin^2(x)}\implies \cfrac{1}{cos^2(x)}\cdot \cfrac{1}{sin^2(x)}\implies sec^2(x)csc^2(x)`