calculus - Why Euler's formula is true?

I need to know why Euler's formula is true? I mean why is the following true:

$$e^{ix} = \cos(x) + i\sin(x)$$

Answer

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: $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + .......$$