simplify sin (8x) using only sin(x) and cos (x)

Answer 1

sin8x = 2sin4xcos4x
= 4sin2xcos2x(cos^2(2x)-sin^2(2x))
= 8sinx*cosx(cos^2(x)-sin^2(x))[(cos^2(x)-sin^2(x)]^2 - 4sin^2(x)cos^2(x)]

and you can expand that in many ways, one of which is, using s=sin(x) c=cos(x):

8sc^7 - 56s^3c^5 + 56s^3c^5 - 8s^7c