How do we derive the expansion for sin 3x?

I think by expansion you mean sin 3x in terms of sin
x.

We start with the relations sin (x + y) = sin x* cos y +
cos x*sin y and cos (x + y) = cos x* cos y – sin x*sin
y

sin 3x = sin (2x + x) = sin 2x* cos x + cos 2x*sin
x

=> sin (x + x)* cos x + cos (x +x)*sin
x

=> [sin x* cos x + cos x*sin x]* cos x + [cos x*
cos x – sin x*sin x]*sin x

=> 2*sin x * (cos x) ^2 +
(cos x) ^2 * sin x – (sin x) ^3

=> 3*sin x (cos x)
^2 – (sin x) ^3

now (cos x) ^2 + (sin x) ^2 = 1 or (cos x)
^2 = 1 – (sin x) ^2

=> 3* sin x*(1 – (sin x) ^2) -
(sin x) ^3

=> 3* sin x – 3(sin x) ^2) - (sin x)
^3

=> 3*sin x – 4*(sin x)
^3

Therefore we get sin 3x = 3*sin x – 4*(sin
x) ^3