$\phi$-Thue-Morse sequences and infinite products

In this article we introduce a new approach to compute infinite products defined by automatic sequences involving the Thue-Morse sequence. As examples, for any positive integers $q$ and $r$ such that $0 \leq r \leq q-1$, we find infinitely many couples of rational functions $R(x)$ and $S(n)$ such that $$\prod_{n=0}^{\infty}R(n)^{\frac{1+a_n}{2}}S(n)^{\frac{1-a_n}{2}}=2cos(\frac{2r+1}{2q}\pi),$$ where $(a_n)_{n \in \mathbf{N}}$ is the Thue-Morse sequence beginning with $a_0=1,a_1= -1$.