Question

Choose the correct option and justify your choice:

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

• A. $\cos 60°$
• B. $\sin 60°$
• C. $\tan 60°$
• D. $sin30°$

Answer 1

The correct option is C

$\tan 60°$

Explaining the correct option(c):

We know from the trigonometry table that: $\tan 30°=\frac{1}{\sqrt{3}}$ and $\tan 60°=\sqrt{3}$

Putting this value in the given expression we get:

$$\begin{gathered} \Rightarrow \frac{2\tan 30°}{1-{\tan }^{2}30°}=\frac{2\times \left({\displaystyle \frac{1}{\sqrt{3}}}\right)}{1-{\left(\frac{1}{\sqrt{3}}\right)}^2} \\ \Rightarrow \frac{2\tan 30°}{1-{\tan }^{2}30°}=\frac{{\displaystyle \frac{2}{\sqrt{3}}}}{1-\left(\frac{1}{3}\right)} \\ \Rightarrow \frac{2\tan 30°}{1-{\tan }^{2}30°}=\frac{{\displaystyle \frac{2}{\sqrt{3}}}}{\left(\frac{2}{3}\right)} \\ \Rightarrow \frac{2\tan 30°}{1-{\tan }^{2}30°}=\frac{{\displaystyle 2\times 3}}{\sqrt{3}\times 2} \\ \Rightarrow \frac{2\tan 30°}{1-{\tan }^{2}30°}=\sqrt{3} \end{gathered}$$

So the given expression has value $\sqrt{3}$ and we know that $\tan 60°=\sqrt{3}$

Hence, Given expression is equal to $\tan 60°$.