Trigonometric polynomial with exactly 2N zeroes

Question

I have proved that a trigonometric polynomial of the form $\sum^{N}_{n=-N}c_ne^{inx}$ has at most 2N zeroes with the Fundamental Theorem of Algebra. Now I am asked to find a trigonometric polynomial with exactly 2N zeroes. Here I got stuck, I have no idea how to approach this problem. Could someone help me with this?

Answer 1

Pick your favorite $2N$ complex numbers of modulus $1$, say $e^{ix_1},...e^{ix_{2N}}, x_k \in [0, 2\pi)$ and consider $e^{-iNx}\Pi_{n=1}^{2N}(e^{ix}-e^{ix_n})=\sum_{n=-N}^Nc_ke^{ikx}$

By construction, it is a trigonometric polynomial of degree $2N$ with the roots in the period interval $[0, 2\pi)$ being precisely $x_k$