On decompositions of trigonometric polynomials

Let ${\mathbb{R}}_t[\theta ]$ be the ring generated over $\mathbb{R}$ by $\cos \theta$ and $\sin \theta$, and ${\mathbb{R}}_t(\theta )$ be its quotient field. In this paper we study the ways in which an element p of ${\mathbb{R}}_t[\theta ]$ can be decomposed into a composition of functions of the form $p=R(q),$ where $\mathbb{R}\in \mathbb{R}(x)$ and $q\in {\mathbb{R}}_t(\theta )$. In particular, we describe all possible solutions of the functional equation $R_1(q_1)=R_2(q_2)$, where $R_1,R_2\in \mathbb{R}[x]$ and $q_1,q_2\in {\mathbb{R}}_t[\theta ]\mathrm{.}$