The ring of real trigonometric polynomials

Desvl at 
The ringThroughout we consider the polynomial ring \[R=\mathbb{R}[\cos{x},\sin{x}].\] This ring has a lot of non-trivial properties which give us a good chance to study commutative ring theory.First of all, note it is immediate that \[R \cong \mathbb{R}[X,Y]/(X^2+Y^2-1)\] if the map is given by \(X ……