Let m and M be respectively the minimum and maximum value values of cos 2 x 1 + sin 2 x sin 2 x 1 + cos 2 x…

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

${\mathrm{C}}_1\to {\mathrm{C}}_1+{\mathrm{C}}_2$

$\begin{array}{l}D=\left|\begin{array}{ccc}2& 1+{\sin }^{2}x& \sin 2x\\ 2& {\sin }^{2}x& \sin 2x\\ 1& {\sin }^{2}x& 1+\sin 2x\end{array}\right|\end{array}$

$\begin{array}{l}{R}_1\to R_1-2R_3\text{ and }{R}_2\to R_2-2R_3\end{array}$

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

As range of $\sin 2x\in \left[-1,1\right]$, thus

$m=-3,M=-1$