Question

Solve $2{\sin }^{2}x+{\sin }^{2}2x=2$.

Answer 1

${\sin }^{2}2x=2(1-{\sin }^{2}x)$
$(2\sin x\cos x{)}^2=2{\cos }^{2}x$
or $2{\sin }^{2}x{\cos }^{2}x-{\cos }^{2}x=0$
$\therefore {\cos }^{2}x(2{\sin }^{2}x-1)=0$
or $\cos x=0$ or $\sin x=\pm 1/\sqrt{2}$
$\therefore x=(n+\frac{1}{2})\frac{\pi }{2},n\pi \pm \frac{\pi }{4}$.