Proof for e^ix

[Hyperreality]

How can I proof the identity

e^ix = cos x + i sin x?

[CTS]

Consider the infinite sums:
e^x = 1 + x/1! + x^2/2! + x^3/3! + ...
sin(x) = x/1! - x^3/3! + x^5/5! -x^7/7! + ...
cos(x) = 1 - x^2/2! + x^4/4! -x^6/6! + ...

e^ix = 1 + ix/1! + (ix)^2/2! + (ix)^3/3!...
Notice the patterns with the powers of i and rearrange the terms to see how they relate to sin and cos.

[mathwonk]

or check that both functions satisfy the same de and same initial conditions,

i.e. f'' + f = 0 and f(0) = 1, f'(0) = i.

[hedlund]

How can I proof the identity

e^ix = cos x + i sin x?

You could use the function z = cos(x) + i*sin(x) with z(0)=1 and dz/dx = -sin(x) + i*cos(x) = i*cos(x) + i^2*sin(x) = i*(cos(x)+i*sin(x)) = iz. Which gives dz/dx = iz <=> dz/z = i dx integration gives \int \frac{dz}{z} = \int i \ dx which gives ln(z) + C = ix + D => ln(z) = ix+E => z = e^(ix+E). Now we have that cos(x)+i*sin(x) = e^(ix+E), and with z(0)=1 it gives that e^(i*0+E)=e^E=1 => E=0 which yields e^(ix)=cos(x)+i*sin(x).

Edit: E = D - C

[HallsofIvy]

In other words, there are a number of different proofs, depending on what you already know and how you are defining the different functions.