The sum of all values of $\mathrm{\theta }\in [0,2\mathrm{\pi }]$ satisfying $2{\sin }^2\mathrm{\theta }=\cos 2\mathrm{\theta }\text{ and }2{\cos }^2\mathrm{\theta }=3\sin \mathrm{\theta }$ is  

$\begin{array}{l}2{\sin }^2\mathrm{\theta }=\cos 2\mathrm{\theta }\\ 2{\sin }^2\mathrm{\theta }=1-2{\sin }^2\mathrm{\theta }\\ 4{\sin }^2\mathrm{\theta }=1\\ {\sin }^2\mathrm{\theta }=\frac{1}{4}\\ \sin \mathrm{\theta }=\pm \frac{1}{2}\\ 2{\cos }^2\mathrm{\theta }=3\sin \mathrm{\theta }\\ 2-2{\sin }^2\mathrm{\theta }+3\sin \mathrm{\theta }-2=0\\ (2\sin \mathrm{\theta }-1)(2\sin \mathrm{\theta }-2)=0\\ \sin \mathrm{\theta }=\frac{1}{2}\end{array}$
so common equation which satisfy both equations is $\sin \mathrm{\theta }=\frac{1}{2}$
$\begin{array}{l}\mathrm{\theta }=\frac{\mathrm{\pi }}{6},\frac{5\mathrm{\pi }}{6}(\mathrm{\theta }\in [0,2\mathrm{\pi }\left]\right)\\ \text{ Sum }=\mathrm{\pi }\end{array}$