Question

Consider $x=4{\tan }^{-1}\left(\frac{1}{5}\right),y={\tan }^{-1}\left(\frac{1}{70}\right)$ and $z={\tan }^{-1}\left(\frac{1}{99}\right)$.

What is $x$ equal to?

• A. ${\tan }^{-1}\left(\frac{60}{119}\right)$
• B. ${\tan }^{-1}\left(\frac{120}{119}\right)$
• C. ${\tan }^{-1}\left(\frac{90}{169}\right)$
• D. ${\tan }^{-1}\left(\frac{170}{169}\right)$

Answer 1

The correct option is B ${\tan }^{-1}\left(\frac{120}{119}\right)$

$x=4{\tan }^{-1}\left(\frac{1}{5}\right)=2{\tan }^{-1}\left(\frac{1}{5}\right)+2{\tan }^{-1}\left(\frac{1}{5}\right)=2{\tan }^{-1}\left(\frac{\frac{1}{5}+\frac{1}{5}}{1-\frac{1}{5}\times \frac{1}{5}}\right)=2{\tan }^{-1}\left(\frac{5}{12}\right)={\tan }^{-1}\left(\frac{120}{119}\right)$