# How do you simplify (cos(-x))/(tan(-x))-sin(x)?

Dec 5, 2017

$- \frac{1}{\sin} \left(x\right)$

#### Explanation:

Things to remember
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{\cos \left(- x\right) = \cos \left(x\right)}$

color(white)("XXX"color(blue)(tan(-x)=-tan(x))

$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{\tan \left(x\right) = \sin \frac{x}{\cos} \left(x\right)}$
and
$\textcolor{w h i t e}{\text{XXX}} \textcolor{m a \ge n t a}{{\cos}^{2} \left(x\right) = 1 - {\sin}^{2} \left(x\right)}$

$\frac{\textcolor{red}{\cos \left(- x\right)}}{\textcolor{b l u e}{\tan \left(- x\right)}} - \sin \left(x\right)$

$\textcolor{w h i t e}{\text{XXX}} = \frac{\textcolor{red}{\cos \left(x\right)}}{\textcolor{b l u e}{- \tan \left(x\right)}} - \sin \left(x\right)$

$\textcolor{w h i t e}{\text{XXX}} = \frac{\textcolor{red}{\cos \left(x\right)}}{- \textcolor{g r e e n}{\frac{\sin \left(x\right)}{\cos \left(x\right)}}} - \sin \left(x\right)$

$\textcolor{w h i t e}{\text{XXX}} \frac{- \left(\textcolor{red}{\cos \left(x\right)}\right) \cdot \left(\textcolor{g r e e n}{\cos \left(x\right)}\right)}{\textcolor{g r e e n}{\sin \left(x\right)}} - \sin \left(x\right)$

$\textcolor{w h i t e}{\text{XXX}} = \frac{- \textcolor{m a \ge n t a}{{\cos}^{2} \left(x\right)}}{\textcolor{g r e e n}{\sin \left(x\right)}} - \sin \left(x\right)$

$\textcolor{w h i t e}{\text{XXX}} = \frac{- \left(\textcolor{m a \ge n t a}{1 - {\sin}^{2} \left(x\right)}\right)}{\textcolor{g r e e n}{\sin \left(x\right)}} - \sin \left(x\right)$

$\textcolor{w h i t e}{\text{XXX}} = \frac{{\sin}^{2} \left(x\right) - 1}{\textcolor{g r e e n}{\sin \left(x\right)}} - \frac{{\sin}^{2} \left(x\right)}{\sin} \left(x\right)$

$\textcolor{w h i t e}{\text{XXX}} = - \frac{1}{\sin} \left(x\right)$