2 tan 30°1+tan2 30° is equal to
Question:

$\frac{2 \tan 30^{\circ}}{1+\tan ^{2} 30^{\circ}}$ is equal to

(a) $\sin 60^{\circ}$

(b) $\cos 60^{\circ}$

(c) $\tan 60^{\circ}$

(d) $\sin 30^{\circ}$

Solution:

We have to find the value of the following expression

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

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

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

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

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

$\left[\begin{array}{l}\text { Since } \tan 60^{\circ}=\frac{\sqrt{3}}{2} \\ \text { Sinec } \tan 30^{\circ}=\frac{1}{\sqrt{3}}\end{array}\right]$

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

$=\sin 60^{\circ}$

Hence the correct option is (a)