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Let $\mathbb{R}[\cos(x),\sin(x)]$ be the algebra generated by the cosine and sine functions over the real numbers. Is it true that it is isomorphic to

$$\dfrac{\mathbb{R}[c,s]}{\langle c^2+s^2-1\rangle}$$

where $\langle c^2+s^2-1\rangle$ is the principal ideal generated by $c^2+s^2-1$ in the (free) polynomial algebra $\mathbb{R}[c,s]$?

Analogous question for hyperbolic sine and cosine functions and the relation $C^2-S^2-1$.

Qfwfq
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  • Here's how to ask a good question. Follow these guidelines to get help in this forum. It's particularly important that you share your own work and thoughts on the problem to show that you have made a serious effort by yourself before asking for help, and you're not just trying to get others to solve it for you. – jjagmath May 28 '24 at 01:01
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    Why the closing votes? Care to elaborate? – Qfwfq May 28 '24 at 01:10
  • Oh I see now: there was already an identical question (that, by the way, didn't appear in the immediate list when I composed this one. But I see it now in the list on the right). I'm totally ok with closing it as solved, then. Should I delete this one? – Qfwfq May 28 '24 at 01:13
  • There's even another identical question: https://math.stackexchange.com/questions/1151024/on-the-polynomial-ring-mathbbrx-y-and-the-sine-and-cosine-functions?rq=1 – Qfwfq May 28 '24 at 01:15
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    I didn't vote to close, but those who did probably wanted to see more context (background, effort, etc.) – J. W. Tanner May 28 '24 at 01:49

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