What is tan^{-1}(1/4)+tan^{-1}(3/5) equal to?

Correct Answer is : π/4π/4π/4

Correct Answer is : π/4π/4π/4

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UPSC NDA I 2018 Mathematics Solved Paper

Question : 63 of 120

Marks: +1, -0

What is

tan^{-1}(\frac{1}{4})+tan^{-1}(\frac{3}{5})

equal to?

Solution:

Concept:

Use the formula

\tan^{-1} A + \tan^{-1} B = \tan^{-1} \frac{A + B}{1 - AB}

for

A, B > 0

and

AB < 1

Explanation:

Let

A = \frac{1}{4}

and

B = \frac{3}{5}

First check

AB = \frac{1}{4} \times \frac{3}{5} = \frac{3}{20} < 1

, so the formula applies.

Then compute:

\frac{A + B}{1 - AB} = \frac{\frac{1}{4} + \frac{3}{5}}{1 - \frac{3}{20}}

Find a common denominator:

\frac{1}{4} + \frac{3}{5} = \frac{5}{20} + \frac{12}{20} = \frac{17}{20}

Denominator:

1 - \frac{3}{20} = \frac{20}{20} - \frac{3}{20} = \frac{17}{20}

Thus the fraction becomes

\frac{17/20}{17/20} = 1

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

We know

\tan^{-1}(1) = \frac{\pi}{4}

since

\tan \frac{\pi}{4} = 1

and the principal value is

(-\frac{\pi}{2}, \frac{\pi}{2})

Answer:

\frac{\pi}{4}

, which corresponds to option B.

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