Let $S\left(\theta \right)$ denote the sum of coefficients in the expansion of ${\left(\sqrt{2}-x\sin \theta +x^2\cos \theta \right)}^{2n}.$Maximum value of $S\left(\theta \right)$ is

$S\left(\theta \right)=(\sqrt{2}-\sin \theta +\cos \theta {)}^{2n}$

Maximum possible value of

$\sqrt{2}-\sin \theta +\cos \theta =\sqrt{2}-\sqrt{2}\left(\cos \frac{\pi }{4}\sin \theta -\right.\left.\sin \frac{\pi }{4}\cos \theta \right)$

$\begin{array}{l}=\sqrt{2}-\sqrt{2}\sin (\theta -\pi /4)\\ \text{ is }\sqrt{2}+\sqrt{2}=2\sqrt{2}\\ \end{array}$

$\therefore$ Maximum possible value of $S\left(\theta \right)\text{ is }{\left(2^{3/2}\right)}^{2n}=8^n$