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Choose the correct answer and justify your choice :
$\dfrac{{2\tan {{30}^0}}}{{1 - {{\tan }^2}{{30}^0}}}$.
A.$\sqrt 3 $
B.$\dfrac{1}{{\sqrt 3 }}$
D.$\sqrt 2 $

Last updated date: 24th Jul 2024
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Hint – In order to solve this problem use the formula of trigonometry for $\tan 2\theta $ then put the value of $\theta $. After that we will be getting the tan of an angle. Putting its value will provide you the right answer.

Complete step-by-step answer:
The given equation is $\dfrac{{2\tan {{30}^0}}}{{1 - {{\tan }^2}{{30}^0}}}$ ……(1)
We know the formula of trigonometry as,
$\dfrac{{2\tan \theta }}{{1 - {{\tan }^2}\theta }} = \tan 2\theta $ ……(2)
Let us put the value of $\theta $ as 30 degrees.
Then the LHS of equation number (2) becomes,
$\dfrac{{2\tan {{30}^0}}}{{1 - {{\tan }^2}{{30}^0}}}$ this is the equation which is asked in question.
From equation number (2) we can say that,
$\dfrac{{2\tan {{30}^0}}}{{1 - {{\tan }^2}{{30}^0}}} = \tan 2({30^0})$
$\dfrac{{2\tan {{30}^0}}}{{1 - {{\tan }^2}{{30}^0}}} = \tan {60^0}$ ......(3)
We know the value of $\tan {60^0} = \sqrt 3 $ ……(4)
Using equation (3) and (4) we can say that,
$\dfrac{{2\tan {{30}^0}}}{{1 - {{\tan }^2}{{30}^0}}} = \sqrt 3 $
Hence the value of the asked equation is $\sqrt 3 $.
 So, the correct option is A.

Note – Whenever you face such types of problems use the general formulas of trigonometry to get the right answer. Here we have used the formula of $\tan 2\theta $ since its value is of the same type as that of the equation given in question. Alternatively, we can also put the value of the angle given in the equation then get the value with the help of the value known to us.