## Precalculus (6th Edition) Blitzer

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 $0$.
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 $0$.