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