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

$\sin^2(x) + \cos^2(x) = 1$, so we have:
$$\sin 2x + \cos^2 x + 1 = 0$$
Now, $\cos^2 x ≥ 0$, and $\sin 2x ≥ -1$, so the only solutions are when $\cos^2 x = 0$ and $\sin 2x = -1$.
When $\cos^2 x = 0$, $x = -\frac{\pi}{2} + 2\pi n$ or $x = \frac{\pi}{2} + 2\pi n$. However, $\sin \left( 2(-\frac{\pi}{2}) \right) = 0$ and $\sin \left( 2(\frac{\pi}{2}) \right) = 0$, so there are no solutions.
