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Consider the space of all bounded continuous real-valued functions of $\mathbb{R}$. I am having trouble understanding how to find the closed subalgebra generated by sine and cosine.

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Denote $\mathcal A$ the subalgebra generated by $x\mapsto \cos x$ and $x\mapsto \sin x$. $\mathcal A$ contains all the maps $\cos(nx)$ and $\sin(nx)$ (can be shown by induction and the formulas of $\cos(a+b)$, $\sin(a+b)$) and also the constants (because $\cos^2x+\sin^2x=1$).

If we take a continuous $2\pi$-periodic function then by Stone-Weierstrass theorem it's in the closure of $\operatorname{span}\{1,\cos(kx),\sin(kx),k\in\mathbb N\}$ so the closure of $\mathcal A$ contains all continuous $2\pi$-periodic functions. Conversely, every function in $\mathcal A$ is continuous and $2\pi$-periodic and so is an uniform limit of such functions.

We conclude that the closure of $\mathcal A$ is the space of all continuous $2\pi$-periodic functions.

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