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  4. The number of soultions of 2 tan -1 x = tan -1| x | is

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

Inverse Trigonometric Functions

Solution:

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