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How to prove Euler’s formula: $\exp(i t)=\cos(t)+i\sin(t)$?

I need to know why Euler's formula is true? I mean why is the following true: $$ e^{ix} = \cos(x) + i\sin(x) $$

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Wrong name, wrong formula. –  Did Oct 31 '12 at 7:31
    
You can try plugging $i \theta$ into the Taylor series for $e^x$, and watch as the Taylor series for $\sin$ and $\cos$ appear, as if by magic. I imagine it was quite a shock to Euler when he did this. –  littleO Oct 31 '12 at 7:31
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$\frac{d}{dx} e^{ix}=ie^{ix}\\ \frac{d}{dx} \cos{x}+i\sin{x}=i(\cos{x}+i\sin{x})$ –  Angela Richardson Oct 31 '12 at 7:37
    
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marked as duplicate by Cameron Buie, copper.hat, AD., Did, Hans Lundmark Oct 31 '12 at 8:13

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Hint: Notice $$\sin (x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ..... $$ and $$i\cos (x) = i - i\frac{x^2}{2!} + i\frac{x^4}{4!} - i\frac{x^6}{6!} + .... $$ Now add them and use the fact that $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. You should obtain $e^{ix}$. Also notice: $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ....... $$

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