Question : The value of $\frac{2 \tan 60^{\circ}}{1+\tan ^2 60^{\circ}}$ is:
Option 1: $\cos 60°$
Option 2: $\tan 60°$
Option 3: $\sin 60°$
Option 4: $\sin 30°$
Correct Answer: $\sin 60°$
Solution :
Given, $\frac{2 \tan 60°}{1+\tan ^2 60°}$
$=\frac{\frac{2\sin60°}{\cos60°}}{1+\frac{\sin^260°}{\cos^260°}}$
$=\frac{2\sin60°\times\cos^260°}{\cos60°(\cos^260°+\sin^260°)}$
$=\frac{2\sin60°\cos60°}{1}$ [As $\cos^2\theta+\sin^2\theta=1$]
$=2\times\frac{\sqrt3}{2}\times\frac12$
$=\frac{\sqrt3}{2}=\sin60°$
Hence, the correct answer is $\sin60°$.
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