Proof of Euler's formula

This is a proof of Euler's formula that uses the circular motion

Suppose
P(t)= e^(it)
is a trajectory on the complex plane, using some definition of the exponential (the Taylor series definition for example) it is possible to show that
P'(t)=(e^(it))'= i e^(it) =i P(t)
and
P(0)= e^(i0)= e^(0)=1
In fact any reasonable definition of exponential should have these properties if you want to take these for granted all this becomes an heuristic proof.

then P(t) describes a trajectory in the complex plane that at t = 0 is at 1 and which velocity P(t) is always perpendicular, (90 degrees counter-clockwise) to P(t) and the velocity P(t) also has the same modulus as P(t). Such trajectory can only be the circular movement with radius one, unit velocity, starting at the point 1: cos(t) + i sin(t)!! If you want rigor again use the uniqueness of solution of differential equations the Picard–Lindelöf theorem.

e^(it)= cos(t) + i sin(t).