# How do you prove (1+tan x) / (1+cot x) = 2?

Jul 11, 2015

Let's suppose we were to even go through with this.

$\frac{1 + \tan x}{1 + \left(\frac{1}{\tan} x\right)} \cdot \frac{\tan x}{\tan x}$

$= \frac{\tan x + {\tan}^{2} x}{\tan x + 1}$

$= \tan \frac{x}{\tan x + 1} + {\tan}^{2} \frac{x}{\tan x + 1}$

$= \frac{\tan x + 1}{\tan x + 1} - \cancel{\frac{1}{\tan x + 1}} + \frac{{\tan}^{2} x - 1}{\tan x + 1} + \cancel{\frac{1}{\tan x + 1}}$

$= \frac{\tan x + 1}{\tan x + 1} + \frac{{\tan}^{2} x - 1}{\tan x + 1}$

$= 1 + \frac{\left(\tan x - 1\right) \cancel{\left(\tan x + 1\right)}}{\cancel{\left(\tan x + 1\right)}}$

$= 1 + \tan x - 1$

$= \textcolor{b l u e}{\tan x}$

So clearly, this is not true. This is equal to $\tan x$.