Question

Prove the identity sin4x=2sin2xcos2x

Answer 1

Ok so think of it this way: 
sin4x=sin2(2x), and lets say that 2x=a
By the double angle theorem, we know that sin2a=2sin(a)cos(a)
So substitute 2x for a, and there you go!

Answer 2

There is a trig identity called the sum of 2 angles for sin its
sin(a+b)=sin(a)cos(b)+cos(a)(sin(b)

You will need to use it. So in your question split the 4x in 2 equal parts 2x and 2x
 
sin(4x)=sin(2x+2x)

Now using the expansion above you will get

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

And it will simplify to 

2sin(2x)cos(2x)

I hope this helps you! Good luck :)