My favorite proofs of trancendental identies these days come down to the fact that solutions of linear differential equations with constant coefficients can be expressed in terms of a basis.

Since sine and cosine satisfy f’’=-f, then any solution of that differential equation can be written as f(t)=a cos(t)+ b sin(t). Since e^it is also a solution:

e^it = a cos(t)+b sin(t)

taking the derivative we also have

ie^it = -a sin(t)+b cos(t)

and setting t=0 in both equations says

a = 1, b = i

as required.