Answer
$$\sin4x=4\sin x\cos x\cos2x$$
The equation is an identity, as proved below.
Work Step by Step
$$\sin4x=4\sin x\cos x\cos2x$$
The right side would be examined first, for it is more complex.
$$X=4\sin x\cos x\cos 2x$$
$$X=2\times(2\sin x\cos x)\times\cos2x$$
- From Double-Angle Identity: $2\sin x\cos x=\sin2x$
So we replace $2\sin x\cos x$ with $\sin 2x$.
$$X=2\sin2x\cos2x$$
- Again, we take advantage of the identity $2\sin x\cos x=\sin 2x$, but this time $$2\sin 2x\cos2x=\sin4x$$
Therefore, $$X=\sin4x$$
That means $$\sin4x=4\sin x\cos x\cos2x$$
The equation is an identity as a result.
