Question

Make tan(x)(csc(x)-sin(x)) into 1 trigononometrical value

Answer 1

$\bf tan(\theta)=\cfrac{sin(\theta)}{cos(\theta)}\qquad\qquad csc(\theta)=\cfrac{1}{sin(\theta)} \\\\\\ sin^2(\theta)+cos^2(\theta)=1\implies cos^2(\theta)=1-sin^2(\theta)\\\\ -------------------------------\\\\ tan(x)[cos(x)-sin(x)]\implies \cfrac{sin(x)}{cos(x)}\left[ \cfrac{1}{sin(x)}-sin(x) \right] \\\\\\ \cfrac{sin(x)}{cos(x)}\left[ \cfrac{1-sin^2(x)}{sin(x)}\right]\implies \cfrac{sin(x)}{cos(x)}\left[ \cfrac{cos^2(x)}{sin(x)}\right]\implies cos(x)$