# Question #88b3e

Feb 13, 2018

$\frac{1}{\cos} x$

#### Explanation:

Convert everything to $\sin x$ and $\cos x$:

$\frac{\sin \frac{x}{\cos} x + \cos \frac{x}{\sin} x}{\frac{1}{\sin} x}$

Dividing by $\frac{1}{\sin} x$ is the same as multiplying by the reciprocal $\sin x$

$\left(\sin x\right) \left(\sin \frac{x}{\cos} x + \cos \frac{x}{\sin} x\right)$

Distribute

${\sin}^{2} \frac{x}{\cos} x + \frac{\sin x \cos x}{\sin} x$

Simplify

${\sin}^{2} \frac{x}{\cos} x + \cos \frac{x}{1}$

The common denominator is $\cos x$

${\sin}^{2} \frac{x}{\cos} x + {\cos}^{2} \frac{x}{\cos} x$

$\frac{{\sin}^{2} x + {\cos}^{2} x}{\cos} x$
${\sin}^{2} x + {\cos}^{2} x = 1$ (Pythagorean Identity) Sooooo:
$\frac{{\sin}^{2} x + {\cos}^{2} x}{\cos} x = \frac{1}{\cos} x$
That's your answer in terms of $\cos x$. Most people would convert $\frac{1}{\cos} x$ to $\sec x$, though.