The solutions of the equation

$\left|\begin{array}{ccc}1+\sin ^{2} x & \sin ^{2} x & \sin ^{2} x \\ \cos ^{2} x & 1+\cos ^{2} x & \cos ^{2} x \\ 4 \sin 2 x & 4 \sin 2 x & 1+4 \sin 2 x\end{array}\right|=0$

use $R_{1} \rightarrow R_{1}+R_{2}+R_{3}$

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

$\Rightarrow$

$x=\frac{\pi}{2}+\frac{\pi}{12}, \pi-\frac{\pi}{12}$