# How do you prove (tan(x)-1)/(tan(x)+1)= (1-cot(x))/(1+cot(x))?

Apr 19, 2015

Start off by cross multiplying

$\left(\tan x - 1\right) \left(1 + \cot x\right) = \left(\tan x + 1\right) \left(1 - \cot x\right)$

Expand each side using FOIL

$\tan x + \tan x \cot x - 1 - \cot x$
$= \tan x - \tan x \cot x + 1 - \cot x$

Since $\tan x$ and $\cot x$ are reciprocals

$\tan x \cot x = 1$

Now we can write

$\tan x + 1 - 1 - \cot x = \tan x - 1 + 1 - \cot x$

Simplifying each side

$\tan x - \cot x = \tan x - \cot x$

The right hand side and left hand side are the same

Apr 19, 2015