The total number of solutions of $|\cot \mathrm{x}|=\cot \mathrm{x}+\frac{1}{\sin \mathrm{x}}$, $\mathrm{x}\in [0, 3\mathrm{\pi }]$, is equal to

$|\cot \mathrm{x}|=\cot \mathrm{x}+\frac{1}{\sin \mathrm{x}}$

If $\cot \mathrm{x}>0$

$\Rightarrow \cot \mathrm{x}=\cot \mathrm{x}+\frac{1}{\sin \mathrm{x}}=0$

$\Rightarrow \frac{1}{\sin \mathrm{x}}=0$, which is not possible.

If $\cot \mathrm{x}\le 0$

$\begin{array}{l}\Rightarrow -\cot \mathrm{x}=\cot \mathrm{x}+\frac{1}{\sin \mathrm{x}}\\ \Rightarrow -2\cot \mathrm{x}=\frac{1}{\sin \mathrm{x}}\end{array}$

or   $\cos \mathrm{x}=-\frac{1}{2}$

$\Rightarrow \mathrm{x}=\frac{2\mathrm{\pi }}{3}, \frac{8\mathrm{\pi }}{3}$