Question

$\tan \left[3\, \tan ^{-1}\left(\frac{1}{5}\right),-\frac{\pi}{4}\right]$ is equal to

• A. $-\frac{13 }{ 46}$
• B. $-\frac{11 }{ 46}$
• C. $-\frac{7 }{ 46}$
• D. $-\frac{4 }{ 23}$
• E. $-\frac{ 9}{46 }$

Answer 1

Answer: (E)

$\tan \left\{3 \tan ^{-1}\left(\frac{1}{5}\right)-\frac{\pi}{4}\right\}$
$=\tan \left\{2 \tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{1}{5}\right)-\tan ^{-1}(1)\right\}$
$=\tan \left\{\tan ^{-1}\left(\frac{2 \times \frac{1}{5}}{1-\frac{1}{25}}\right)+\tan ^{-1}\left(\frac{\frac{1}{5}-1}{1+\frac{1}{5}}\right)\right\}$
$=\tan \left\{\tan ^{-1}\left(\frac{2}{5} \times \frac{25}{24}\right)+\tan ^{-1}\left(-\frac{4}{5} \times \frac{5}{6}\right)\right\}$
$=\tan \left\{\tan ^{-1}\left(\frac{5}{12}\right)-\tan ^{-1}\left(\frac{2}{3}\right)\right\}$
$=\tan \left(\tan ^{-1} \frac{\frac{5}{12}-\frac{2}{3}}{1+\frac{5}{18}}\right)$
$=\tan \left\{\tan ^{-1}\left(\frac{-3}{12} \times \frac{18}{23}\right)\right\}$
$=-\frac{9}{46}$