Adding both the given equations, we get,

$2x=3+4\sin 2\theta -\cos 4\theta$
$=3+4\sin 2\theta -\left(1-2{\sin }^{2}2\theta \right)$
$=2{\sin }^{2}2\theta +4\sin 2\theta +2$
$\Rightarrow x={\sin }^{2}2\theta +2\sin 2\theta +1$
$\Rightarrow x={\left(\sin 2\theta +1\right)}^2$
Similarly, $2y=3-\cos 4\theta -4\sin 2\theta$
$2y=3-4\sin 2\theta -\left(1-2{\sin }^{2}2\theta \right)$
$=2{\sin }^{2}2\theta -4\sin 2\theta +2$
$\Rightarrow y={\sin }^{2}2\theta -2\sin 2\theta +1$
$\Rightarrow y={\left(\sin 2\theta -1\right)}^2$
$\sqrt{x}=\left|\sin 2\theta +1\right|=1+\sin 2\theta$
$\sqrt{y}=\left|\sin 2\theta -1\right|=1-\sin 2\theta$
$\Rightarrow \sqrt{x}+\sqrt{y}=2$