trigonometry - Verify the following trigonometric identity.

Verify the following trig identity: $$\sin(3\theta)-\sin\theta = 2\cos(2\theta)\sin\theta$$

Here is my work so far.

$\sin(3\theta)-\sin\theta = 2\cos(2\theta)\sin\theta$

LHS:$$\sin(\theta+2\theta)-\sin\theta$$
$$\sin\theta \cos(2\theta)+\sin(2\theta)\cos\theta-\sin\theta$$
$$\sin\theta \cos(2\theta)+(2\sin\theta \cos\theta)\cos\theta-\sin\theta$$

Where do I go from here? I think I should leave the first term in the line above as is, and try and manipulate the second two terms to equal $\sin\theta \cos(2\theta)$, then the LHS will add together to equal $2\cos(2\theta)\sin\theta$, and the identity will be verified. How do you suggest I get there?

Any hints or advice would be appreciated.

EDIT:

I ended up verifying this identity using the identity frank000 mentioned in comments. Thanks to everyone for the input, it was all very helpful.

Answer

hint: also use $\sin(\theta) = \sin(2\theta - \theta) = \sin(2\theta)\cos(\theta)-\sin(\theta)\cos(2\theta)$