Answer
The expression on the left-side is equal to the expression on the right-side.
Work Step by Step
Let us consider the given expression on the left side $\tan \left( x+\frac{3\pi }{4} \right)$.
Now, it can be simplified by using the identity:
$\tan \left( \theta +\phi \right)=\frac{\tan \theta +\tan \phi }{1-\tan \theta \tan \phi }$
$\begin{align}
& \tan \left( x+\frac{3\pi }{4} \right)=\frac{\tan x+\tan \frac{3\pi }{4}}{1-\tan x.\tan \frac{3\pi }{4}} \\
& =\frac{\tan x+\left( -1 \right)}{1-\tan x.\left( -1 \right)} \\
& =\frac{\tan x-1}{1+\tan x}
\end{align}$
Therefore, the expression can be further simplified, where $\theta =x$ and $\phi =\frac{3\pi }{4}$.
Hence, the given expression on the left side is equal to the expression on the right side.
$\tan \left( x+\frac{3\pi }{4} \right)=\frac{\tan x-1}{1+\tan x}$.
