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