The value of { ^{ - 1}}≤ft( {3/5} ) + { ^{ - 1}}≤ft( {1/7} ) is equal to

Correct Answer is : π4π /44π

Correct Answer is : π4π /44π

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UPSC NDA 2 2017 Mathematics Solved Paper

Question : 35 of 120

Marks: +1, -0

The value of

{\sin ^{ - 1}}\left( {\frac{3}{5}} \right) + {\tan ^{ - 1}}\left( {\frac{1}{7}} \right)

is equal to

Solution:

Concept:

Convert the inverse sine term into an inverse tangent using a right triangle, then apply the sum formula for inverse tangents.

Explanation:

Let

\theta = \sin^{-1}\left(\frac{3}{5}\right)

Then

\sin\theta = \frac{3}{5}

Using the Pythagorean identity,

\cos\theta = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}

Thus

\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{3/5}{4/5} = \frac{3}{4}

, so

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

Therefore,

\sin^{-1}\left(\frac{3}{5}\right) = \tan^{-1}\left(\frac{3}{4}\right)

Now the expression becomes

\tan^{-1}\left(\frac{3}{4}\right) + \tan^{-1}\left(\frac{1}{7}\right)

Apply the identity

\tan^{-1}A + \tan^{-1}B = \tan^{-1}\left(\frac{A+B}{1-AB}\right)

Set

A = \frac{3}{4}

and

B = \frac{1}{7}

Then

\frac{A+B}{1-AB} = \frac{\frac{3}{4} + \frac{1}{7}}{1 - \frac{3}{4} \cdot \frac{1}{7}} = \frac{\frac{21+4}{28}}{1 - \frac{3}{28}} = \frac{\frac{25}{28}}{\frac{25}{28}} = 1

\tan^{-1}\left(\frac{3}{4}\right) + \tan^{-1}\left(\frac{1}{7}\right) = \tan^{-1}(1) = \frac{\pi}{4}

Answer:

\frac{\pi}{4}

(Option B)

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