Answer
The exact value of $\frac{2\tan \frac{\pi }{12}}{1-{{\tan }^{2}}\frac{\pi }{12}}$ is $\frac{\sqrt{3}}{3}$.
Work Step by Step
Recall the given expression.
$\tan 2\theta =\frac{2\tan \theta }{1-{{\tan }^{2}}\theta }$
Apply the given expression.
$\begin{align}
& \frac{2\tan \frac{\pi }{12}}{1-{{\tan }^{2}}\frac{\pi }{12}}=\tan 2\left( \frac{\pi }{12} \right) \\
& =\tan \left( \frac{\pi }{6} \right) \\
& =\frac{\sqrt{3}}{3}
\end{align}$
Therefore, the exact value of $\frac{2\tan \frac{\pi }{12}}{1-{{\tan }^{2}}\frac{\pi }{12}}$ is $\frac{\sqrt{3}}{3}$.
