Let f x = sin 2 x - 2 + cos 2 x cos 2 x 2 + sin 2 x cos 2 x cos 2 x sin 2 x cos 2 x 1 + cos 2 x ,   x…

Given,

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

$=\left|\begin{array}{ccc}-2& -2& 0\\ 2& 0& -1\\ {\sin }^{2}x& {\cos }^{2}x& 1+\cos 2x\end{array}\right|\left(\begin{array}{l}{R}_1\to R_1-R_2\\ \& R_2\to R_2-R_3\end{array}\right)$

$=-2\left({\cos }^{2}x\right)+2\left(2+2\cos 2x+{\sin }^{2}x\right)$

$=4+4\cos 2x-2\left({\cos }^{2}x-{\sin }^{2}x\right)$

$\Rightarrow f\left(x\right)=4+2\cos 2x$

We know, $-1\le \cos 2x\le 1$

So, $f{\left(x\right)}_{\max }=4+2=6$