What's the value of cos(x) that verifies the following equality

Question

The equality is:

$\ 2 \sin^2 2x - 2\sin x \sin 3x = 4 \cos x + \cos 2x $

Answer 1

Use that $$\sin(2x)=2\sin(x)\cos(x)$$ $$\cos(2x)=\cos^2(x)-\sin(x)^2$$ $$\sin(3x)=4\sin(x)\cos^2(x)-\sin(x)$$ So you will get $$2(1-\cos^2(x))-4\cos(x)-2\cos^2(x)+1=0$$ and let $$t=\cos(x)$$

Answer 2

Hint

$$4\cos x+\cos2x=1-\cos4x-(\cos2x-\cos4x)$$

Now replace $\cos2x$ with $2\cos^2x-1$ to form a quadratic equation in $\cos x$

Used http://mathworld.wolfram.com/WernerFormulas.html