Let $\zeta$ be an $n$-th root of unity and let $\chi:=\zeta+\zeta^{-1}$. Then $\zeta^k+\zeta^{-k}=P_k(\chi)$ where $P_k\in\Bbb{Z}[X]$ is a polynomial not depending on $n$. For example we have \begin{eqnarray*} \zeta^2+\zeta^{-2}&=&\chi^2-2, \qquad&\text{ so }&\qquad P_2&=&X^2-2,\\ \zeta^3+\zeta^{-3}&=&\chi^3-3\chi, \qquad&\text{ so }&\qquad P_3&=&X^3-3X,\\ \zeta^4+\zeta^{-4}&=&\chi^4-4\chi^2+2, \qquad&\text{ so }&\qquad P_4&=&X^4-4X^2+2,\\ \zeta^5+\zeta^{-5}&=&\chi^5-5\chi^3+5\chi, \qquad&\text{ so }&\qquad P_5&=&X^5-5X^3+5,\\ \zeta^6+\zeta^{-6}&=&\chi^6-6\chi^4+9\chi^2+18, \qquad&\text{ so }&\qquad P_6&=&X^6-6X^4+9X^2+18. \end{eqnarray*} It isn't hard to see that $P_{ab}=P_a\circ P_b=P_b\circ P_a$ for all positive integers $a$ and $b$, and that we have a recurrence relation $$P_a=X^a-\sum_{i=1}\binom{a}{i}P_{a-2i},$$ where we take the convention that $P_k=0$ for all $k<0$, and $P_0=1$. My question is:
Is there a simple explicit expression for $P_k$?
As I said, Chebyshev polynomials are exactly what you said.