# Chapter 5 - Section 5.2 - Sum and Difference Formulas - Exercise Set - Page 669: 28

The expression $\frac{\tan 50{}^\circ -\tan 20{}^\circ }{1+\tan 50{}^\circ \tan 20{}^\circ }$ is written as $\tan 30{}^\circ$ and the exact value of $\tan 30{}^\circ$ 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( 50{}^\circ -20{}^\circ \right)=\frac{\tan 50{}^\circ -\tan 20{}^\circ }{1+\tan 50{}^\circ \tan 20{}^\circ } \\ & \tan \left( 30{}^\circ \right)=\frac{\tan 50{}^\circ -\tan 20{}^\circ }{1+\tan 50{}^\circ \tan 20{}^\circ } \end{align} Therefore, the expression $\frac{\tan 50{}^\circ -\tan 20{}^\circ }{1+\tan 50{}^\circ \tan 20{}^\circ }$ is equivalent to $\tan 30{}^\circ$. From the knowledge of trigonometric ratios defined for tangent of an angle, the exact value of $\tan 30{}^\circ$ is $\frac{1}{\sqrt{3}}$.

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