I want to express $$f(x) = \sin^3x$$ in terms of its fourier series using the fourier series of $sin2x$ and $cos^2 x$ between the interval $[-\pi, \pi]$ for a period of $2\pi$. Respectively here is what I found:
$$\sin2x = \sum^{\infty}_1b_n \sin(nx)$$
$$\cos^2x = \frac{a_0}{2} + \sum^{\infty}_1a_n cos(nx)$$
But from what I have searched online, $sin^3x$ requires an expression in terms of $sin3x$, so I'm unable to use this fact with my requirements of $cosx$ and sin$2x$