How can I simplify $\cos(2x) + \frac{1}{2}\sin(4x) - \frac{1}{2}\cos(4x)$ into a single expression?

Question

How can I simplify the following expression into a single trigonometric expression?

$$\cos(2x) + \frac{1}{2}\sin(4x) - \frac{1}{2}\cos(4x)$$

Answer 1

The closest answers I have are $$2 \cos 2x \sin x (\cos x + \sin x)$$ and $$\sqrt {2} \cos 2x \sin x (\sin x + \pi/4)$$ using the following identities:

$$\cos 4x = 2 \cos^2 2x -1$$ $$\sin 4x = 2 \sin 2x \cos 2x$$...and somewhere along the line, you can also use $$\cos 2x = \cos^2 x - \sin^2 x$$ in order to eliminate one of the factors.

Answer 2

Hint: $\cos2x=\cos^2x-\sin^2x$ and $\sin2x=2 \sin x \cos x$