Question

The value of $4{\tan }^{-1}\frac{1}{5}-{\tan }^{-1}\frac{1}{70}+{\tan }^{-1}\frac{1}{99}=$

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. None of these

Answer 1

The correct option is C $\frac{\pi }{4}$

$4{\tan }^{-1}\frac{1}{5}-{\tan }^{-1}\frac{1}{70}+{\tan }^{-1}\frac{1}{99}=2{\tan }^{-1}\left[\frac{\frac{2}{5}}{1-\frac{1}{25}}\right]-{\tan }^{-1}\frac{1}{70}+{\tan }^{-1}\frac{1}{99}=2{\tan }^{-1}\left(\frac{5}{12}\right)-{\tan }^{-1}\frac{1}{70}+{\tan }^{-1}\frac{1}{99}={\tan }^{-1}\left[\frac{\frac{5}{6}}{1-\frac{25}{144}}\right]-{\tan }^{-1}\frac{1}{70}+{\tan }^{-1}\frac{1}{99}$
$={\tan }^{-1}\left(\frac{120}{119}\right)-{\tan }^{-1}\frac{1}{70}+{\tan }^{-1}\frac{1}{99}={\tan }^{-1}\left(\frac{120}{119}\right)+{\tan }^{-1}\left[\frac{\frac{1}{99}-\frac{1}{70}}{1+\frac{1}{99}.\frac{1}{70}}\right]$

$={\tan }^{-1}\left(\frac{120}{119}\right)+{\tan }^{-1}(-\frac{29}{6931})={\tan }^{-1}\frac{120}{119}-{\tan }^{-1}\frac{29}{6931}={\tan }^{-1}\frac{120}{119}-{\tan }^{-1}\frac{1}{239}$

$={\tan }^{-1}\left[\frac{\frac{120}{119}-\frac{1}{239}}{1+\frac{120}{119}\times \frac{1}{239}}\right]={\tan }^{-1}\left(\frac{120\times 239-119}{120+119\times 239}\right)={\tan }^{-1}\left(1\right)=\frac{\pi }{4}$