Question

The value of $\tan \left[2\,\tan^{-1}\frac{1}{5}-\frac{\pi}{4}\right]$ is

• A. 0
• B. 1
• C. $-\frac{7}{17}$
• D. none of these.

Answer 1

Answer: (C)

$-\frac{7}{17}$

Answer 2

$2\,tan^{-1} \frac{1}{5} = tan^{-1} \frac{1}{5} +tan^{-1} \frac{1}{5}$ $= tan^{-1} \frac{\frac{1}{5} +\frac{1}{5}}{1-\frac{1}{5}\cdot\frac{1}{5}}$ $= tan^{-1} \frac{\frac{2}{5}}{\frac{24}{25}}$ $= tan^{-1} \frac{5}{12}$ $tan \left(2tan^{-1} \frac{1}{5} -\frac{\pi}{4}\right) = tan \left(tan^{-1} \frac{5}{12} -\frac{\pi}{4}\right)$ $=\frac{ tan \left(tan^{-1} \frac{5}{12}\right) - tan \frac{\pi}{4}}{1+tan\left(tan^{-1} \frac{5}{12}\right) tan \frac{\pi}{4}}$ $=\frac{ \frac{5}{12}-1}{1+\frac{5}{12}\cdot1}$ $= -\frac{\frac{7}{12}}{\frac{17}{12}}$ $= -\frac{7}{17}$