## Precalculus (6th Edition) Blitzer

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 $\frac{1}{\sqrt{3}}$.
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 $\frac{1}{\sqrt{3}}$.