On decompositions of trigonometric polynomials

Let R_t[θ] be the ring generated over R by cosθ and sinθ, and R_t(θ) be its quotient field. In this paper we study the ways in which an element p of R_t[θ] can be decomposed into a composition of functions of the form p = R ◦ q, where R ∈ R(x) and q ∈ R_t(θ). In particular, we describe all possible solutions of the functional equation R₁ ◦ q₁ = R₂ ◦ q₂, where R₁,R₂ ∈ R[x] and q₁, q₂ ∈ R_t[θ].