Answer
The expression $\frac{\tan \frac{\pi }{5}-\tan \frac{\pi }{30}}{1+\tan \frac{\pi }{5}\tan \frac{\pi }{30}}$ is written as $\tan \frac{\pi }{6}$ and the exact value of $\tan \frac{\pi }{6}$ is $\frac{1}{\sqrt{3}}$.
Work Step by Step
Use the difference formula of tangent and rewrite the expression as the difference of angles to obtain the tangent of the angle as,
$\begin{align}
& \tan \left( \frac{\pi }{5}-\frac{\pi }{30} \right)=\frac{\tan \frac{\pi }{5}-\tan \frac{\pi }{30}}{1+\tan \frac{\pi }{5}\tan \frac{\pi }{30}} \\
& \tan \left( \frac{\pi }{6} \right)=\frac{\tan \frac{\pi }{5}-\tan \frac{\pi }{30}}{1+\tan \frac{\pi }{5}\tan \frac{\pi }{30}}
\end{align}$
Therefore, the expression $\frac{\tan \frac{\pi }{5}-\tan \frac{\pi }{30}}{1+\tan \frac{\pi }{5}\tan \frac{\pi }{30}}$ is equivalent to $\tan \frac{\pi }{6}$.
From the knowledge of trigonometric ratios defined for tangent of an angle, the exact value of $\tan \frac{\pi }{6}$ is $\frac{1}{\sqrt{3}}$.
