How to solve $\sin^2(x)+\sin2x+2\cos^2(x)=0$ How do you solve $\sin^2(x)+\sin2x+2\cos^2(x)=0$?
I have been able to rewrite it as $(\sin(x)+\cos(x))^2+\cos^2(x)=0$. Not obviously useful, I think

Rushabh's answer is perfect, though I would do it slightly differently. Note that $\sin 2x = 2\sin x\cos x$, so your equation can be rewritten as
$$ \sin^2 x + 2\sin x\cos x + 2\cos^2 x = 0.$$
If you treat $\sin x$ and $\cos x$ as independent variables $u$ and $v$, this becomes the quadratic equation
$$ u^2 + 2uv + 2v^2 = 0, $$
where it becomes clear there are no solutions, as the discriminant is $\triangle = 2^2-4\times 2 = -4<0$.
