Q65 / 125 · BITSAT 2014 · Mathematics
Find the value of $\tan \left[2 \tan ^{-1} \frac{1}{5}-\frac{\pi}{4}\right]$
- A$-1 / 3$
- Correct answerB$-7 / 17$
- C$-1 / 2$
- D$-1 / 4$
Solution
$\begin{aligned} & 2 \tan ^{-1} \frac{1}{5}=\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{5} \\ &=\tan ^{-1} \frac{\frac{1}{5}+\frac{1}{5}}{1-\frac{1}{5} \cdot \frac{1}{5}}=\tan ^{-1} \frac{2 / 5}{24 / 25}=\tan ^{-1} \frac{5}{12} \\ &=\tan \left(2 \tan ^{-1} \frac{1}{5}-\frac{\pi}{4}\right)=\tan \left(\tan ^{-1} \frac{5}{12}-\frac{\pi}{4}\right) \\ &=\frac{\tan \left(\tan ^{-1} \frac{5}{12}\right)-\tan \frac{\pi}{4}}{1+\tan \left(\tan ^{-1} \frac{5}{12}\right) \tan \frac{\pi}{4}}=\frac{-\frac{7}{12}}{\frac{17}{12}}=-\frac{7}{17} \end{aligned}$
Hence, the correct answer is option: 'B'.