Question

Simplify 1-cot(x)/tan(x)-1

Answer 1

$\bf \cfrac{1-cot(x)}{tan(x)-1}\qquad \begin{cases} cot(\theta)=\cfrac{cos(\theta)}{sin(\theta)} \\\\ tan(\theta)=\cfrac{sin(\theta)}{cos(\theta)} \end{cases}\qquad thus \\\\\\ \cfrac{1-\frac{cos(x)}{sin(x)}}{\frac{sin(x)}{cos(x)}-1}\implies \cfrac{\frac{sin(x)-cos(x)}{sin(x)}}{\frac{sin(x)-cos(x)}{cos(x)}}\\\\ -----------------------------\\\\ recall\implies \cfrac{\frac{a}{b}}{\frac{c}{{{ d}}}}\implies \cfrac{a}{b}\cdot \cfrac{{{ d}}}{c}\qquad thus\\\\ -----------------------------$

$\bf \cfrac{\frac{sin(x)-cos(x)}{sin(x)}}{\frac{sin(x)-cos(x)}{cos(x)}}\implies \cfrac{sin(x)-cos(x)}{sin(x)}\cdot \cfrac{cos(x)}{sin(x)-cos(x)}\implies \boxed{?}$