Verify the identity: sin 4x = 2 sin 2x cos 2x

Sin 2x = 2 sin x cos x

that is one of the double angle formulas

4x = sin 2(2x)

then it immediately follows that

sin 2(2x) = 2 sin 2x cos 2x

Verify the identity: sin 4x = 2 sin 2x cos 2x
We use a trigonometric identity as follows:

sin(a+b)=sin(a)cos(b)+cos(a)(sin(b)

We start with

sin(4x)=sin(2x+2x)
sin(2x+2x)=sin(2x)×cos(2x)+cos(2x)×sin(2x)

Then, it will simplify to

2sin(2x)cos(2x)

Hope this answers the question.