Answer
See the explanation given below.
Work Step by Step
Let us consider the left side:
$\tan x+\tan y $
We know that, $\tan x=\frac{\sin x}{\cos x}$.
It implies:
$\begin{align}
& \tan x+\tan y=\frac{\sin x}{\cos x}+\frac{\sin y}{\cos y} \\
& =\frac{\sin x\cos y+\sin y\cos x}{\cos x\cos y}
\end{align}$
By using the identity $\sin \left( x+y \right)=\sin x\cos y+\sin y\cos x $, we obtain:
$\tan x+\tan y=\frac{\sin \left( x+y \right)}{\cos x\cos y}$
Thus, $\tan x+\tan y=\frac{\sin \left( x+y \right)}{\cos x\cos y}$
