Pythagoras says that $\cos^2 \theta + \mathrm{sin}^2\theta = 1$ for all real $\theta$.

**(Vague) Question.** Is this the *only* relationship between the functions $\cos$ and $\sin$?

More precisely:

Let $\langle \cos,\sin\rangle$ denote the intersection of all subalgebras of the $\mathbb{R}$-algebra of all functions $\mathbb{R} \rightarrow \mathbb{R}$ containing $\{\cos,\sin\}$. (By default, all my algebras are unital, associative and commutative.) Let $A$ denote the $\mathbb{R}$-algebra presented by the generators $\{c,s\}$ and the relation $c^2+s^2=1$. There is a unique $\mathbb{R}$-algebra homomorphism $\varphi : A \rightarrow \langle \cos,\sin\rangle$ given as follows. $$\varphi(c) = \cos, \,\,\varphi(s) = \sin$$

We know that $\varphi$ is surjective.

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> **Question.** Is $\varphi$ injective?
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So consider $f \in A$. Then $f = \sum\_{i,j : \mathbb{N}}a\_{ij}s^ic^i$ for certain choices of $a\_{ij} : \mathbb{R}$. Now suppose $\varphi(f)=0$. We want to show that $f=0$. Ideas, anyone?