Solve for `x :` `tan^(-1)(2+x)+tan^(-1)(2-x)=tan^(-1)2/3` , where `x<-sqrt(3)` or, `x >sqrt(3)`

Answer

`x= pm 3`

Explanation

As we know that
$$
\begin{aligned}
&\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right) \\
&\therefore \tan ^{-1}(2+x)+\tan ^{-1}(2-x)=\tan ^{-1} \frac{2}{3} \\
&\Rightarrow \tan ^{-1}\left(\frac{2+x+2-x}{1-(2+x) \times(2-x)}\right)=\tan ^{-1} \frac{2}{3} \\
&\Rightarrow \frac{4}{1-4+x^{2}}=\frac{2}{3} \\
&\Rightarrow-6+2 x^{2}=12 \\
...