How does this factorise?

Question

I was completing a question in a booklet and marked it, but I don't understand how this expression:

$$-2\sin 2x + 2\cos x = 0$$

Turns into this:

$-2(\sin x\cos x + \sin x\cos x) + 2\cos x = 0$

The rest of the question made sense to me

Answer 1

It is a standard formula: $$ \sin 2x = 2\sin x\cdot \cos x .$$

It can fairly simply be derived from the addition theorem for sin: $$\sin (x+y) = \sin x\cos y+\sin y \cos x.$$

Just put $y=x$.

Likewise, $$\sin 3x = \sin 2x\cos x +\sin x \cos 2x =...$$

Answer 2

The substitution used is $\sin 2x=\sin x\cos x+\cos x\sin x,$ which follows from the basic property of the $\sin$ function $$\sin(a+b)=\sin a\cos b+\cos b\sin a,$$ by taking $a=b=x.$