In order to understand the proof, you need to go quite deep into mathematics. But here’s the idea.
The exponential function exp is the unique function that satisfy the differential equation
| u’ = u
| u(0) = 1
(u’ is the derivative of u). If you don’t know what a differential equation is, it is “simply” an equation of which the solution is a function.
The important thing is that this is true everywhere in the complex plane (so, not only for real numbers). Let’s see what happens with the function v(z) = u(iz) when we derive it.
v’(z) = i*u’(iz) = i * u(iz) = i*v(z) (because u’ = u)
v’’(z) = i² * u’(iz) =-v(z) (because i² = -1 and again, u’ = u).
More over we have
v(0) = 1 and v’(0) = i
We thus derived a differential equation for v, it has to satisfy
| v’’ = -v
| v(0) = 1
| v’(0) = i
It’s “easy” to see that the unique solution to this equation is
v(z) = cos(z) + i sin(z)
Since v(z) is also equal to u(iz) = exp(iz), we must have
exp(iz) = cos(z) + i sin(z)