The maximum value of f x = sin 2 x 1 + cos 2 x cos 2 x 1 + sin 2 x cos 2 x cos 2 x sin 2 x cos 2 x sin 2 x ,…

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

$C_1\to C_1+C_2$

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

$R_1\to R_1-R_2$

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

Expanding w.r.t. $R_1$

$=-\left(2\sin 2x-\cos 2x\right)$

So, $f\left(x\right)=\cos 2x-2\sin 2x$

${\overline{)f\left(x\right)}}_{\max }=\sqrt{1+4}=\sqrt{5}$