Sum of all the solutions in  $[0,4\pi ]$ of the equation  $\tan x+\cot x+1=\cos (x+\frac{\pi }{4})$  is

$\tan x+\cot x+1=\cos (x+\frac{\pi }{4})$

$\Rightarrow \tan x+\cot x=\cos (x+\frac{\pi }{4})-1$

But $\tan x+\cot x\le -2$ and  $\cos (x+\frac{\pi }{4})-1\ge -2$

$\Rightarrow$ equality holds when both are $-2$

$\Rightarrow \cos (x+\frac{\pi }{4})=-1$

$\Rightarrow x+\frac{\pi }{4}=(2n+1)\pi ,m\in Z$

$\Rightarrow x=\frac{3\pi }{4}$  or  $\frac{11\pi }{4}$

Sum $\frac{3\pi }{4}+\frac{11\pi }{4}=\frac{7\pi }{2}$.