$\frac{2 \tan 30^{\circ}}{1+\tan ^{2} 30^{\circ}}$ is equal to
(a) $\sin 60^{\circ}$
(b) $\cos 60^{\circ}$
(c) $\tan 60^{\circ}$
(d) $\sin 30^{\circ}$
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)
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