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

Feb 19, 2016

It is not an identity and thus, it can't be proven.

#### Explanation:

You can't prove it since it's not an identity.

Let's transform the left side using

$\tan x = \sin \frac{x}{\cos} x$, $\text{ } \cot x = \cos \frac{x}{\sin} x$

We have

$\frac{1 + \tan x}{1 + \cot x} = \frac{1 + \sin \frac{x}{\cos} x}{1 + \cos \frac{x}{\sin} x}$

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

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

$= \frac{\cos x + \sin x}{\cos} x \cdot \sin \frac{x}{\sin x + \cos x}$

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

$= \sin \frac{x}{\cos} x$

$= \tan x$

However, $\tan x = 2$ is certainly not true for all $x$.

Thus, your equation is not an identity and can't be proven.