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When proving them the "modern" way (from first principles) it seems impossible to get around proving the identities $$\lim_{x\to 0}\frac{\sin x}{x} = 1$$ and the related $\cos$ limit. This itself requires some geometric reasoning and uses the definition of radians that we have today.

But surely people knew that $(\sin \theta)'=\cos \theta$ and so forth long before limits came about. Is this just obvious? How would they have justified it?

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