$\tan (x+\frac{\pi }{4})+\tan (x-\frac{\pi }{4})=2\sqrt{3}$ $\Rightarrow \frac{\tan x+1}{1-\tan x}+\frac{\tan x-1}{1+\tan x}=2\sqrt{3}$ $\Rightarrow \frac{{{(1+\tan x)}^{2}}-{{(1-\tan x)}^{2}}}{1-{{\tan }^{2}}x}=2\sqrt{3}$ $\Rightarrow \frac{4\tan x}{1-{{\tan }^{2}}x}=2\sqrt{3}\Rightarrow \frac{2\tan x}{1-{{\tan }^{2}}x}=\sqrt{3}$ $\Rightarrow \tan 2x=\tan \frac{\pi }{3}\Rightarrow 2x=k\pi +\frac{\pi }{3}\Rightarrow x=\frac{k\pi }{2}+\frac{\pi }{6}$