# How do I simplify the trigonometric expression? cos(x)/sec(x) + tan(x)

Feb 7, 2018

${\cos}^{2} x + \tan x$

#### Explanation:

$\cos \frac{x}{\sec} x + \tan x$

Identity:

$\textcolor{red}{\boldsymbol{\sec x = \frac{1}{\cos} x}}$

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

${\cos}^{2} x + \tan x$

Feb 7, 2018

It can be simplified to $1 - \sin x$

#### Explanation:

I will assume that the question asks $\frac{\cos x}{\sec x + \tan x}$

Using our fundamental identities of $\sec x = \frac{1}{\cos} x$ and $\tan x = \sin \frac{x}{\cos} x$, we will get:

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

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

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

We know that ${\sin}^{2} x + {\cos}^{2} x = 1$, therefore:

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

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

$1 - \sin x$

Hopefully this helps!