Consider the following equality:
$$a_1\sin(x) + a_2\sin^2(x) + ... +a_n\sin^n(x) = a_1\sin(x) + a_2\sin(2x) + ... + a_n\sin(nx)$$
where $a_1, a_2\dots, a_n \in \mathbb{C}, \;n \in \mathbb{N}$.
This equality clearly holds for $n = 1$ and for any $a_1 \in \mathbb{C}$. The trivial equality ($a_1 = a_2 = \cdots = a_n = 0$) also holds.
- Do there exists any other solutions? If so, are there infinitely many of such solutions?
- Do there exist other solutions if we constrain the domain of coefficients to $\mathbb{Q}$, $\mathbb{R}$ or $\mathbb{Z}$?
With $z:=e^{ix}$, you want to establish the identity
$$\sum_k a_k\left(\frac{z-z^{-1}}{2i}\right)^k=\sum_k a_k\frac{kz-(kz)^{-1}}{2i}.$$
By identification of the polynomial coefficients, only $a_1$ can be nonzero.