Projective Modules over the Ring of Trigonometric Functions

Let $R = \mathbb{R}[ \cos x, \sin x]$ and consider the ideal $\langle 1 - \cos x, \sin x\rangle$. Is this ideal a projective module over $R$ ?

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The ring $\mathbb{R}[\cos x, \sin x]$ is isomorphic to $\mathbb{R}[X,Y]/(X^2+Y^2-1)$ which is known as being a Dedekind domain, so all ideals are projective.

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I'm interested to know where I can find an explanation of why it is Dedekind. Thanks! – rschwieb Nov 13 '12 at 17:15
It's an interesting example that I had not seen in detail before. I imagine all commutative algebraists must be familiar with it! – rschwieb Nov 13 '12 at 22:03
@rschwieb, you can see it here, math.colorado.edu/~ravi1033/notes/pre08/pdf/… I already read that before this post, expecting some straight forward solution, but on your reply I found that it is again the same complex manipulations :-) – Ram Nov 14 '12 at 2:10