Question

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

Answer 1

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

$\Rightarrow 2{\sin }^{2}x+4{\sin }^{2}x{\cos }^{2}x=2$

$\Rightarrow 2{\sin }^{2}x(1+2{\cos }^{2}x)=2$

$\Rightarrow {\sin }^{2}x(2{\cos }^{2}x+{\sin }^{2}x)=1$

$\Rightarrow {\sin }^{2}x=2{\cos }^{2}x+{\sin }^{2}x=1$

$\Rightarrow 2{\cos }^{2}x=0$

$\Rightarrow x=\frac{\pi }{2},\frac{3\pi }{2},\frac{5\pi }{2}$

$\Rightarrow x=(2n+1)\frac{\pi }{2}$