do an analytic continuation from exp x to all complex values; the same with cos x and sin x; then is easy to verify(by power series) that (1) e^(iz) = cos z + isen z; doing z = pi u have the result.
You can think that (1) is heuristic based that both classes of function (exp x and cos x + isen x) are given by the same functional equation viz:
f( x + y ) = f( x )*f( y );
sorry for my possible lack of analysis :)