The value of ≤ft[2 ^{-1} 1/5-π/4] is

Correct Answer is : −717-7/1717−7

Correct Answer is : −717-7/1717−7

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MHTCET 5 May 2025 Shift 2 Paper

Question : 148 of 150

Marks: +1, -0

The value of

\tan \left[2 \tan ^{-1} \frac{1}{5}-\frac{\pi}{4}\right]

Solution:

Concept:

Use the double-angle identity for inverse tangent and the formula for the difference of two inverse tangents.

Explanation:

First, apply

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

with

x = \frac{1}{5}

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

Now rewrite

\frac{\pi}{4}

\tan^{-1}1

\tan\left(2\tan^{-1}\frac{1}{5}-\frac{\pi}{4}\right) = \tan\left[\tan^{-1}\left(\frac{5}{12}\right) - \tan^{-1}1\right]

Use the identity

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

= \tan\left[\tan^{-1}\left(\frac{\frac{5}{12}-1}{1+\frac{5}{12}}\right)\right] = \tan\left[\tan^{-1}\left(\frac{-\frac{7}{12}}{\frac{17}{12}}\right)\right]

= \tan\left[\tan^{-1}\left(-\frac{7}{17}\right)\right] = -\frac{7}{17}

Answer:

-\frac{7}{17}

, which is option C.

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