Question

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

Answer 1

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

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

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

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

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

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

${\sin }^{2}x{\cos }^{2}x=1-{\sin }^{2}x$

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

${\sin }^{2}x{\cos }^{2}x-{\cos }^{2}x=0$

$({\sin }^{2}x-1){\cos }^{2}x=0$

$\Rightarrow {\sin }^{2}x=1,{\cos }^{2}x=0$

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