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Given the definition of cos function through the exponential function, how can we prove rigorously that for real values of $z$ $$\cos(z)=\operatorname{Re}(\exp(iz))$$?

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How about looking at the power series of $\cos z$? –  sos440 Nov 15 '12 at 1:58
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Prove first that $\mathrm{Re}(z) = (z + \overline{z})/2$ for any complex number $z$. –  GEdgar Nov 15 '12 at 2:04

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Notice that $\overline{e^z} =e^{\overline{z}}$. Hence for $z\in \mathbb{R}$, $$\text{cos}(z) = \frac{e^{iz}+e^{-iz}}{2} = \frac{e^{iz}+e^{\overline{iz}}}{2} = \frac{e^{iz}+\overline{e^{iz}}}{2} = \text{Re}(e^{iz})$$

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