The solutions of the equation

$\left|\begin{array}{ccc}1+{\sin }^2\mathrm{x}& {\sin }^2\mathrm{x}& {\sin }^2\mathrm{x}\\ {\cos }^2\mathrm{x}& 1+{\cos }^2\mathrm{x}& {\cos }^2\mathrm{x}\\ 4 \sin 2\mathrm{x}& 4 \sin 2\mathrm{x}& 1+4 \sin 2\mathrm{x}\end{array}\right|=0, (0<\mathrm{x}<\mathrm{\pi }) \mathrm{are}$

$$\begin{gathered} {\mathrm{R}}_1\to {\mathrm{R}}_1+{\mathrm{R}}_2+{\mathrm{R}}_3 \\ \Rightarrow (2+4\sin 2\mathrm{x}) \left|\begin{array}{ccc}1 & 1 & 1\\ {\cos }^2\mathrm{x}& 1+{\cos }^2\mathrm{x}& {\cos }^2\mathrm{x}\\ 4\sin 2\mathrm{x}& 4\sin 2\mathrm{x}& 1+4\sin 2\mathrm{x}\end{array}\right|=0 \\ \sin 2\mathrm{x}=\frac{-1}{2} \\ 2\mathrm{x}=\mathrm{\pi }+\frac{\mathrm{\pi }}{6}, 2\mathrm{\pi }-\frac{\mathrm{\pi }}{6} \\ \mathrm{x}=\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{12}, \mathrm{\pi }-\frac{\mathrm{\pi }}{12} \\ =\frac{7\mathrm{\pi }}{12}, \frac{11\mathrm{\pi }}{12} \end{gathered}$$