Question : If $A=30^{\circ}$, then find the value of $\frac{(2 \tan A)}{\left(1-\tan^2 A\right)}$.
Option 1: $4 \sqrt{3}$
Option 2: $\frac{3}{\sqrt{3}}$
Option 3: $3$
Option 4: $2 \sqrt{3}$
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Correct Answer: $\frac{3}{\sqrt{3}}$
Solution :
Given: $A=30^{\circ}$
And we know, $\tan 30° = \frac{1}{\sqrt3}$
So, $\frac{(2 \tan A)}{\left(1-\tan ^2 A\right)}$
$= \frac{2\tan 30°}{1-\tan^2 30°}$
$= \frac{2\times \frac{1}{\sqrt3}}{1-\frac{1}{3}} = \frac{2\times 3}{2\times \sqrt3} = \frac{3}{\sqrt 3}$
Hence, the correct answer is $\frac{3}{\sqrt3}$.
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