2nd order homogeneous differential equations with constant coeficients

I was learning about how to solve linear 2nd order homogeneous differential equations with constant coeficients in which the caracteristic equation has two complex solutions(indeed conjugate complex numbers, it couldn't be otherwise!)

My problem with that is at the very end of it! when extracting the real part of the complex number I found it hard to know why the hell does it always have a sin part when e^it is cos(t)-sin(t)

can anyone help me plz?