Question

How do you simplify $\frac{(1+\tan x)}{(1-\tan x)}$ ?

Answer 1

Simplify the required term.

We know that, $\tan \left(\frac{\mathrm{\pi }}{4}\right)=1$

So, $\frac{(1+\tan x)}{(1-\tan x)}$ can be written as $\frac{\tan \left({\displaystyle \frac{\mathrm{\pi }}{4}}\right)+\tan x}{1-\tan \left(\frac{\mathrm{\pi }}{4}\right)\tan x}$.

Since, $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$

$\Rightarrow \tan (\frac{\mathrm{\pi }}{4}+x)$$=\frac{\tan \left({\displaystyle \frac{\mathrm{\pi }}{4}}\right)+\tan x}{1-\tan \left(\frac{\mathrm{\pi }}{4}\right)\tan x}$

Hence, $\frac{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{x}\mathbf{)}}{\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{x}\mathbf{)}}$can be simplified as $\mathit{t}\mathit{a}\mathit{n}\left(\frac{\pi }{4}+x\right)$.