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)

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