You don't even need to think specifically about exp(ix) unless you're doing calculus and taking derivatives or antiderivatives. Just the fact that there's SOME base of exponentiation such that cos is the even component of exponentiation with this base and i * sin is the odd component of exponentiation with this base is enough for all these trig laws.

Incidentally, we can derive [cos(x)]^2 + [sin(x)]^2 = 1 from the fact that cos and i * sin are even and odd components of exponentiation with some base, like so: 1 = exp(ix) * exp(-ix) = (cos(x) + i * sin(x)) * (cos(x) - i * sin(x)) = [cos(x)]^2 + [sin(x)]^2.