Find the principal solutions of the trigonometric equation $\cos x-\sin x+\sin 2x+3\cos2x+1=0$ I am unable to simplify the expression. If I simplify the double angles, it leaves me with a nasty expression, 
$\cos x-\sin x+2\sin x\cos x+6\cos^2 x-2=0$. What do I do next. Some hints, please. Also, is there some elegant solution? Thanks!

Observe:
$$
\cos x-\sin x+2\sin x\cos x+6\cos^2 x-2=(\sin x)(2\cos x-1)+(2\cos x-1)(3\cos x+2)=(\sin x+3\cos x-2)(2\cos x-1).
$$
Therefore, $\cos x=\frac{1}{2}$ is a solution.
The solutions to $\sin x+3\cos x+2=0$ are less pleasant, but one can solve $\sin x=-2-3\cos x$, square both sides and solve, using the quadratic formula for $\cos x$.
