Question

Choose the correct option and justify your choice:
$\frac{2\tan {30}^{\circ }}{1+{\tan }^2{30}^{\circ }}=$

• A. $\sin {60}^{\circ }$
• B. $\cos {60}^{\circ }$
• C. $\tan {60}^{\circ }$
• D. $\sin {30}^{\circ }$

Answer 1

The correct option is A $\sin {60}^{\circ }$

Given trigonometric expression is $\frac{2\tan {30}^{\circ }}{1+{\tan }^2{30}^{\circ }}$

We know that, $\tan {30}^{\circ }=\frac{1}{\sqrt{3}}$

$\therefore \frac{2\tan {30}^{\circ }}{1+{\tan }^2{30}^{\circ }}$

$=\frac{2\left(\frac{1}{\sqrt{3}}\right)}{1+{\left(\frac{1}{\sqrt{3}}\right)}^2}$

$=\frac{\frac{2}{\sqrt{3}}}{1+\frac{1}{3}}$

$=\frac{\frac{2}{\sqrt{3}}}{\frac{4}{3}}$

$=\frac{6}{4\sqrt{3}}$

$=\frac{\sqrt{3}}{2}$

We know that, $\sin {60}^{\circ }=\frac{\sqrt{3}}{2}$

$\therefore \frac{2\tan {30}^{\circ }}{1+{\tan }^2{30}^{\circ }}=\sin {60}^{\circ }$

Hence, option A is correct.