If you are really serious when you asked why are those two expression equal it is because if you plug in any x-value from the domain the left hand side will always equal the right hand side. Such an equation is called an identity.
Now if meant how to prove this identity that is very different.
I would write the left hand side in terms of sin(x) and cos(x). You can do that by using the double angle formulas and the sum rules.
For example: sin(4x) = sin (2*2x) = (2)sin(2x)*cos(2x) = (2)*2sin(x)cos(x)*(cos^2(x)-sin^2(x))
