Question

If $tan^{-1}x = \frac {\pi}{4} - tan^{-1} (\frac {1}{3})$ then $x$ is

• A. $\frac {1}{3}$
• B. $\frac {1}{2}$
• C. $\frac {1}{4}$
• D. $\frac {1}{6}$

Answer 1

Answer: (B)

Given equation $\tan ^{-1} x=\frac{\pi}{4}-\tan ^{-1}\left(\frac{1}{3}\right)$
$\Rightarrow \tan ^{-1} x=\tan ^{-1}(1)-\tan ^{-1}\left(\frac{1}{3}\right)$
$\left(\because \frac{\pi}{4}=\tan ^{-1}(1)\right)$
$\Rightarrow \tan ^{-1} x=\tan ^{-1}\left(\frac{1-\frac{1}{3}}{1+\frac{1}{3}}\right)$
$\left[\because \tan ^{-1} x-\tan ^{-1} y= \tan ^{-1}\left(\frac{x-y}{1+x y}\right)\right]$
$\Rightarrow \tan ^{-1} x=\tan ^{-1}\left(\frac{2 / 3}{4 / 3}\right)$
$\Rightarrow \tan ^{-1} x=\tan ^{-1}(1 / 2)$
$\Rightarrow x=1 / 2$