Question

The value of $2{\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{7}$ is

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

Answer 1

Answer: (A)

$\frac{\pi }{4}$

Given, ${\tan }^{-1}\left(\frac{1}{3}\right)+{\tan }^{-1}\left(\frac{1}{3}\right)+{\tan }^{-1}\left(\frac{1}{7}\right)$
$={\tan }^{-1}\left(\frac{1}{3}\right)+{\tan }^{-1}\left(\frac{7+3}{21-1}\right)$
$={\tan }^{-1}\left(\frac{1}{3}\right)+{\tan }^{-1}\left(\frac{10}{20}\right)$
$={\tan }^{-1}\left(\frac{1}{3}\right)+{\tan }^{-1}\left(\frac{1}{2}\right)$
$={\tan }^{-1}\left(\frac{5}{6-1}\right)$
$={\tan }^{-1}\left(1\right)$
$=\frac{\pi }{4}$