Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 5 - Section 5.2 - Sum and Difference Formulas - Exercise Set - Page 669: 32


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 $0$.
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