Math Question: How do you prove this?

Question

Prove sin(3x)=3sin(x)-4(sin(x)^3)

I got to sin(3x)=sin(2x+x)=sin(2x)cos(x)+cos(2x)s...

Then you use this identity cos2x=1-2sin(x)^2 to get

sin(2x)cos(x)+sin(x)-2sin(x)^3

I can't get the left side of the plus sign to turn into 2sin(x)-sin(x)^3

I think it has to do with the same identity as above, but I can't get it.

Answer 1

Sin(3x)

= sin(2x+x)

= sin2x cosx + cos2x sinx

= 2sinx cos(x)^2 + [cos(x)^2 - sin(x)^2] sinx

= 3sinx cos(x)^2 - sin(x)^3

= 3sinx [1 - sin(x)^2] - sin(x)^3

= 3sinx - 3sin(x)^2 - sin(x)^3

= 3sinx - 4sin(x)^3