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-(cos(x))/(sin(x))}{(sin(x))/(cos(x))-1}\implies \cfrac{(sin(x)-cos(x))/(sin(x))}{(sin(x)-cos(x))/(cos(x))}\\\\ -----------------------------\\\\ recall\implies \cfrac{(a)/(b)}{\frac{c}{{{ d}}}}\implies \cfrac{a}{b}\cdot \cfrac{{{ d}}}{c}\qquad thus\\\\ -----------------------------`


`\bf \cfrac{(sin(x)-cos(x))/(sin(x))}{(sin(x)-cos(x))/(cos(x))}\implies \cfrac{sin(x)-cos(x)}{sin(x)}\cdot \cfrac{cos(x)}{sin(x)-cos(x)}\implies \boxed{?}`