$\cot 2x+\cot x=\tan x+3\Rightarrow \cot 2x+\cot x-\tan x=3$  $\Rightarrow \cot 2x+2\cot 2x=3\Rightarrow 3\cot 2x=3\Rightarrow \cot 2x=1\Rightarrow 2x=k\pi +\frac{\pi }{4}\Rightarrow x=\frac{k\pi }{2}+\frac{\pi }{8}\begin{matrix}    {} & \left( k\in z \right)  \\ \end{matrix}$