Question

Choose the correct option and justify your choice:

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

• A. $\sin 60°$
• B. $\cos 60°$
• C. $\tan 60°$
• D. $\sin 30°$

Answer 1

The correct option is A

$\sin 60°$

Explaining the correct option(a):

We know from the trigonometry table that: $\tan 30°=\frac{1}{\sqrt{3}}$ and $\sin 60°=\frac{\sqrt{3}}{2}$

Putting this value in the given expression we get:

$$\begin{gathered} \Rightarrow \frac{2\tan 30°}{1+{\tan }^{2}30°}=\frac{2\times \left({\displaystyle \frac{1}{\sqrt{3}}}\right)}{1+{\left(\frac{1}{\sqrt{3}}\right)}^2} \\ \Rightarrow \frac{2\tan 30°}{1+{\tan }^{2}30°}=\frac{{\displaystyle \frac{2}{\sqrt{3}}}}{1+\left(\frac{1}{3}\right)} \\ \Rightarrow \frac{2\tan 30°}{1+{\tan }^{2}30°}=\frac{{\displaystyle \frac{2}{\sqrt{3}}}}{\left(\frac{4}{3}\right)} \\ \Rightarrow \frac{2\tan 30°}{1+{\tan }^{2}30°}=\frac{{\displaystyle 2\times 3}}{\sqrt{3}\times 4} \\ \Rightarrow \frac{2\tan 30°}{1+{\tan }^{2}30°}=\frac{\sqrt{3}}{2} \end{gathered}$$

So the given expression has value $\frac{\sqrt{3}}{2}$ and we know that $\sin 60°=\frac{\sqrt{3}}{2}$

Hence, the given expression is equal to $\sin 60°$.

The explanation for the incorrect option:

The value of $\cos 60°=\frac{1}{2}\ne \frac{\sqrt{3}}{2}$

The value of $\tan 60°=\sqrt{3}\ne \frac{\sqrt{3}}{2}$

The value of $\sin 60°=\frac{1}{2}\ne \frac{\sqrt{3}}{2}$

Therefore, the correct option is (A).