$\text{Number of roots of the equation }\left|sinxcosx\right|+\sqrt{2+ta{n}^{2}x+co{t}^{2}x} =\sqrt{3}, x\in \left[0,4\mathrm{\pi }\right] is$

$$\begin{gathered} \left|sinxcosx\right|+\left|tanx+cotx\right|=\sqrt{3} \\ \left|sinxcosx\right|+\frac{1}{\left|sinxcosx\right|}=\sqrt{3} \\ But \left|sinxcosx\right|+\frac{1}{\left|sinxcosx\right|} \ge 2 \\ Hence, no solution. \\ ie; the number of real solutions is zero. \end{gathered}$$