Question

Evaluate $\frac{2\tan 30°}{1-{\tan }^{2}30°}$

Answer 1

Given $\frac{2\tan 30°}{1-{\tan }^{2}30°}$

Where $\tan 30°=\frac{1}{\sqrt{3}}$

So, $\frac{2\tan 30°}{1-{\tan }^{2}30°}$

$$\begin{gathered} =\frac{2\left({\displaystyle \frac{1}{\sqrt{3}}}\right)}{1-{\left(\frac{1}{\sqrt{3}}\right)}^2} \\ =\frac{{\displaystyle \frac{2}{\sqrt{3}}}}{1-\left({\displaystyle \frac{1^2}{{\left(\sqrt{3}\right)}^2}}\right)} \\ =\frac{{\displaystyle \frac{2}{\sqrt{3}}}}{1-{\displaystyle \frac{1}{3}}} \\ =\frac{{\displaystyle \frac{2}{\sqrt{3}}}}{{\displaystyle \frac{3-1}{3}}} \\ =\frac{{\displaystyle \frac{2}{\sqrt{3}}}}{{\displaystyle \frac{2}{3}}} \\ =\frac{2}{\sqrt{3}}\times \frac{3}{2} \\ =\frac{3}{\sqrt{3}} \\ =\frac{\sqrt{3}\sqrt{3}}{\sqrt{3}} \\ =\sqrt{3} \end{gathered}$$

Hence, $\frac{2\tan 30°}{1-{\tan }^{2}30°}=\sqrt{3}$.