Answer
The expression $\frac{\tan \frac{\pi }{5}+\tan \frac{4\pi }{5}}{1-\tan \frac{\pi }{5}\tan \frac{4\pi }{5}}$ is written as $\tan \pi $ and the exact value of $\tan \pi $ is $0$.
Work Step by Step
Use the sum formula of tangent and rewrite the expression as the sum of angles to obtain the tangent of the angle as,
$\begin{align}
& \tan \left( \frac{\pi }{5}+\frac{4\pi }{5} \right)=\frac{\tan \frac{\pi }{5}+\tan \frac{4\pi }{5}}{1-\tan \frac{\pi }{5}\tan \frac{4\pi }{5}} \\
& \tan \left( \pi \right)=\frac{\tan \frac{\pi }{5}+\tan \frac{4\pi }{5}}{1-\tan \frac{\pi }{5}\tan \frac{4\pi }{5}}
\end{align}$
Therefore, the expression $\frac{\tan \frac{\pi }{5}+\tan \frac{4\pi }{5}}{1-\tan \frac{\pi }{5}\tan \frac{4\pi }{5}}$ is equivalent to $\tan \pi $.
From the knowledge of trigonometric ratios defined for tangent of an angle, the exact value of $\tan \pi $ is $0$.
