Question

Simplify $\tan [2{\tan }^{-1}\frac{1}{5}-\frac{\pi }{4}]=\cdots$

• A. $\frac{5}{4}$
• B. $\frac{5}{16}$
• C. $\frac{-7}{17}$
• D. $\frac{7}{17}$

Answer 1

The correct option is C $\frac{-7}{17}$

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

$=\tan ({\tan }^{-1}\frac{\frac{2}{5}}{1-\frac{1}{25}}-\frac{\pi }{4})$

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

$=\tan ({\tan }^{-1}\frac{5}{12}-{\tan }^{-1}(1\left)\right)$

$=\tan \left({\tan }^{-1}\right(\frac{\frac{5}{12}-1}{1+\frac{5}{12}}\left)\right)$

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

$=-\tan \left({\tan }^{-1}\right(\frac{7}{17}\left)\right)=\frac{-7}{17}$.