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UPSC NDA 2 2019 Mathematics Solved Paper

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Question : 45 of 120

Marks: +1, -0

What is

tan{2tan^{-1}(\frac{1}{3})}

equal to?

$2/3$
$3/4$
$3/8$
$1/9$

Solution:

Concept:

Use the double-angle formula for inverse tangent:

2 \tan^{-1} x = \tan^{-1}\left(\frac{2x}{1-x^{2}}\right)

, valid for

-1 < x < 1

.

Explanation:

Let

x = \frac{1}{3}

. Since

-1 < \frac{1}{3} < 1

, the formula applies.

Compute

\frac{2x}{1-x^{2}} = \frac{2 \times \frac{1}{3}}{1 - \left(\frac{1}{3}\right)^{2}} = \frac{2/3}{1 - 1/9} = \frac{2/3}{8/9} = \frac{2}{3} \times \frac{9}{8} = \frac{3}{4}

.

Thus,

2 \tan^{-1}\left(\frac{1}{3}\right) = \tan^{-1}\left(\frac{3}{4}\right)

.

Now,

\tan\left(2\tan^{-1}\left(\frac{1}{3}\right)\right) = \tan\left(\tan^{-1}\left(\frac{3}{4}\right)\right) = \frac{3}{4}

.

Answer:

\frac{3}{4}

, which corresponds to option B.

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