$\tan \left( x+\frac{\pi }{4} \right)+\tan \left( x-\frac{\pi }{4} \right)=2\sqrt{3}\Rightarrow \frac{\tan x+1}{1-\tan x}+\frac{\tan x-1}{1+\tan x}=2\sqrt{3}$  $\Rightarrow \frac{{{\left( 1+\tan x \right)}^{2}}-{{\left( 1-\tan x \right)}^{2}}}{1-{{\tan }^{2}}x}=2\sqrt{3}$  $\Rightarrow \frac{4\tan x}{1-{{\tan }^{2}}x}=2\sqrt{3}\Rightarrow \frac{2\tan x}{1-{{\tan }^{2}}x}=\sqrt{3}$  $\Rightarrow \tan 2x=\tan \frac{\pi }{3}\Rightarrow 2x=k\pi +\frac{\pi }{3}\begin{matrix}    {} & \left( k\in z \right)\Rightarrow x=\frac{k\pi }{2}+\frac{\pi }{6}  \\ \end{matrix}$