Linear span of some trigonometric functions

Question

Decide, if the function $\cos(3x)$ is in the linear span of this set $ \{1, \sin(x), \sin^2(x), \sin^3(x) \} $ (in the vector space of all functions).

Well, I should probably somehow express $\cos(3x)$ as the linear combination of that set, but I wasn't successful.

Answer 1

Suppose it could be done then for some a,b,c,d cos(3x)=a+bsin(x)+csin^2(x)+dsin^3(x). Put x=0 to obtain a=1 . again put x=pi to get a=-1.So we achieve a contradiction.