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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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    See: http://pundit.pratt.duke.edu/wiki/Complex_Numbers and http://en.wikipedia.org/wiki/Complex_number2012-10-31

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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!} + ....... $