# Question #603e4

Jan 22, 2017

You need to multiply out the brackets and use ${\cos}^{2} x + {\sin}^{2} x = 1$ and $\tan x = \sin \frac{x}{\cos} x$

#### Explanation:

$\left(1 - \cos x\right) \left(1 + \frac{1}{\cos x}\right)$
Multiply out the brackets:
$= 1 \times 1 - 1 \cdot \cos x + 1 \times \frac{1}{\cos x} - \frac{\cos x}{\cos x}$
$= 1 - \cos x + \frac{1}{\cos x} - 1$
Cancel the $1$'s:
$= - \cos x + \frac{1}{\cos x}$
Put over a common denominator:
$= \frac{- {\cos}^{2} x + 1}{\cos x}$
Use Pythagoras: ${\cos}^{2} x + {\sin}^{2} x = 1$
$= {\sin}^{2} \frac{x}{\cos} x$

$= \frac{\sin x \times \sin x}{\cos} x$
Use $\tan x = \sin \frac{x}{\cos} x$
$= \tan x \sin x$