Question

The number of soultions of $2{\tan }^{-1}x={\tan }^{-1}|x|$ is

Answer 1

Answer: 2

$2{\tan }^{-1}x={\tan }^{-1}|x|$
$\Rightarrow {\tan }^{-1}\left(\frac{x}{1-x^2}\right)={\tan }^{-1}|x|$
$\Rightarrow \frac{x}{1-x^2}=|x|$
case (i) $x\ge 0,\frac{x}{1-x^2}-x=0$
$\Rightarrow x=0$
case (ii) $x\le 0,\frac{+x}{1-x^2}+x=0$
$\Rightarrow x=02-x^2=0$
$\Rightarrow x^2=2$
$\Rightarrow x=-\sqrt{2}$
$\therefore$ required solutions are $x=0,-\sqrt{2}$