This expression:

$$\cos(4x)\cos(3x) - 4\sin(x) \sin(3x)\cos(x)\cos(2x)$$

I have struggled for hours trying to simplify without success.

Is it legitimate to simplify the right side using the Double Angle identity thusly?

$$\cos(4x)\cos(3x) - \sin(4x)\sin(3x)\cos(2x)$$

That is, if $\sin(2x) = 2\sin(x)\cos(x)$ is it legitimate to say $\sin(4x) = 4\sin(x)\cos(x)$?

But even if I get this far, I want to say it looks like a cosine sum identity

$$\cos(4x + 3x) = \cos(4x)\cos(3x) - \sin(4x)\sin(3x)$$

but I cannot see what to do with that pesky $\cos(2x)$ on the right side.