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