The value of {{tan}}≤ft( {2{{ }^{ - 1}}1/5 - π /4} ) is

Correct Answer is : −717- {7}{{17}}−177

Correct Answer is : −717- {7}{{17}}−177

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UPSC NDA II 2015 Mathematics Paper

Question : 3 of 120

Marks: +1, -0

The value of

{\rm{tan}}\left( {2{{\tan }^{ - 1}}\frac{1}{5} - \frac{\pi }{4}} \right)

Solution:

Concept:

Use inverse trigonometric identities to simplify the expression.

Explanation:

First, apply the formula

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

with

x = \frac{1}{5}

Compute:

\frac{2 \times \frac{1}{5}}{1 - \left(\frac{1}{5}\right)^2} = \frac{\frac{2}{5}}{\frac{24}{25}} = \frac{5}{12}

Thus,

2\tan^{-1}\frac{1}{5} = \tan^{-1}\frac{5}{12}

Next, rewrite

\frac{\pi}{4}

\tan^{-1}1

The given expression becomes

\tan\left(\tan^{-1}\frac{5}{12} - \tan^{-1}1\right)

Use the identity

\tan^{-1}a - \tan^{-1}b = \tan^{-1}\left(\frac{a-b}{1+ab}\right)

Here

a = \frac{5}{12}

b = 1

, so

\frac{\frac{5}{12} - 1}{1 + \frac{5}{12} \times 1} = \frac{-\frac{7}{12}}{\frac{17}{12}} = -\frac{7}{17}

Therefore,

\tan\left(2\tan^{-1}\frac{1}{5} - \frac{\pi}{4}\right) = \tan\left(\tan^{-1}\left(-\frac{7}{17}\right)\right) = -\frac{7}{17}

Answer:

-\frac{7}{17}

(Option A)

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