Answer
See the full explanation below.
Work Step by Step
$\tan \,2\alpha $
By using the identity of trigonometry,
$\tan \,\left( \alpha +\beta \right)=\frac{\tan \,\alpha +\tan \,\beta }{1-\tan \,\alpha \tan \,\beta }$
Now, the above expression can be further simplified as,
$\begin{align}
& \tan \,2\alpha =\tan \,\left( \alpha +\alpha \right) \\
& =\frac{\tan \,\alpha +\tan \,\alpha }{1-\tan \,\alpha \tan \,\alpha } \\
& =\frac{2\tan \,\alpha }{1-{{\tan }^{2}}\,\alpha }
\end{align}$
Thus, the left side of the given expression is equal to the right side, which is,
$\tan \,2\alpha =\frac{2\tan \,\alpha }{1-{{\tan }^{2}}\,\alpha }$.
