Question

Verify the identity.
sec(x) − cos(x) = tan(x) sin(x)

Answer 1

`\bf sec(x) -cos(x) = tan(x) sin(x)\\\\ -----------------------------\\\\ sec(\theta)=\cfrac{1}{cos(\theta)}\\\\ -----------------------------\\\\ thus \\\\\\ \cfrac{1}{cos(x)}-cos(x)\implies \cfrac{1-cos^2(x)}{cos(x)}\\\\ -----------------------------\\\\ recall\implies sin^2(\theta)+cos^2(\theta)=1\implies sin^2(\theta)=1-cos^2(\theta)\\\\ -----------------------------\\\\ \cfrac{sin^2(x)}{cos(x)}\implies \cfrac{sin(x)}{cos(x)}\cdot sin(x)\implies tan(x)sin(x)`