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BITSAT 2014

Exam: BITSAT
Year: 2014
Subject: Mathematics
Chapter: Inverse Trigonometric Functions

BITSAT 2014 · Mathematics · Inverse Trigonometric Functions

Find the value of [2 ^ -1 (1)/(5)-(π)/(4)]

Find the value of

\tan \left[2 \tan ^{-1}

$\frac{1}{5}-\frac{\pi}{4}\right]

Options

A $-1 / 3$
B $-7 / 17$
C $-1 / 2$
D $-1 / 4$

Correct answer

B. $-7 / 17$

Step-by-step solution

\begin{aligned} & 2 \tan ^{-1} \frac{1}{5}=\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{5} \\ &=\tan ^{-1} \frac{\frac{1}{5}+\frac{1}{5}}{1-\frac{1}{5} \cdot \frac{1}{5}}=\tan ^{-1} \frac{2 / 5}{24 / 25}=\tan ^{-1} \frac{5}{12} \\ &=\tan \left(2 \tan ^{-1} \frac{1}{5}-\frac{\pi}{4}\right)=\tan \left(\tan ^{-1} \frac{5}{12}-\frac{\pi}{4}\right) \\ &=\frac{\tan \left(\tan ^{-1} \frac{5}{12}\right)-\tan \frac{\pi}{4}}{1+\tan \left(\tan ^{-1} \frac{5}{12}\right) \tan \frac{\pi}{4}}=\frac{-\frac{7}{12}}{\frac{17}{12}}=-\frac{7}{17} \end{aligned}

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