Solve: $3\sin{2x}+4\cos{2x}-2\cos{x}+6\sin{x}-6=0$ 
Solve:   $3\sin{2x}+4\cos{2x}-2\cos{x}+6\sin{x}-6=0$

My Try
$6\sin{x}\cos{x}+4(\cos^2{x}-\sin^2{x})-2\cos{x}+6\sin{x}-6=0$
I have expanded the equation, But I cannot proceed further, Any hint would be appreciated. Thank you!

Like Barry Cipra,
replacing $\sin x=s,\cos x=c,\cos2x=2c^2-1,\sin2x=2sc$
$$3(2sc)+4(3c^2-1)-2c+6s-6=0$$
$$\iff4c^2-c(1-3s)+3s-5=0$$
$$\implies c=\cdots=-1,\dfrac{5-3s}4$$
If $\cos x=c=-1, x=(2n+1)\pi$
Else $4\cos x+3\sin x=5$
Use Solving trigonometric equations of the form $a\sin x + b\cos x = c$
See also: Find $4\cos\theta-3\sin\theta$, given that $4\sin \theta +3\cos \theta = 5$
