# How do you simplify cotx / (cscx)?

Dec 5, 2015

$\cos x$

#### Explanation:

Recall:
$1. \cot x = \frac{1}{\tan} x$ or $\cos \frac{x}{\sin} x$
$2. \csc x = \frac{1}{\sin} x$

Substitute your reciprocal and quotient identities into the equation:

$\cot \frac{x}{\csc} x$

$= \frac{\cos \frac{x}{\sin} x}{\frac{1}{\sin} x} \Leftarrow$ use $\cot x = \cos \frac{x}{\sin} x$ instead of $\frac{1}{\tan} x$

$= \cos \frac{x}{\sin} x \div \frac{1}{\sin} x$

$= \cos \frac{x}{\sin} x \cdot \sin \frac{x}{1}$

$= \cos \frac{x}{\textcolor{red}{\cancel{\textcolor{b l a c k}{\sin}}}} x \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{\sin}}} \frac{x}{1}$

$= \cos x$