Number of solutions of the equation $4\left({\cos }^{2}2\mathrm{x}+\cos 2\mathrm{x}+1\right)+\tan \mathrm{x}(\tan \mathrm{x}-2\sqrt{3})=0$ in $[0, 2\mathrm{\pi }]$ is

$\begin{array}{l}\left(4{\cos }^{2}2\mathrm{x}+4\cos 2\mathrm{x}+1\right)+\left({\tan }^2\mathrm{x}-2\sqrt{3}\tan \mathrm{x}+3\right)=0\\ \Rightarrow (2\cos 2\mathrm{x}+1{)}^2+(\tan \mathrm{x}-\sqrt{3}{)}^2=0\\ \Rightarrow \cos 2\mathrm{x}=-\frac{1}{2}\text{ and }\tan \mathrm{x}=\sqrt{3}\\ \Rightarrow 2{\cos }^2\mathrm{x}-1=-\frac{1}{2}\text{ and }\tan \mathrm{x}=\sqrt{3}\\ \Rightarrow \cos \mathrm{x}=\pm \frac{1}{2}\text{ and }\tan \mathrm{x}=\sqrt{3}\\ \Rightarrow \mathrm{x}=\frac{\mathrm{\pi }}{3},\frac{4\mathrm{\pi }}{3}\end{array}$

$\therefore$ Number of solutions of equation = 2