Complex roots in recurrence relation

Question

I was reading brilliant wiki on recurrence relations. It says, if the characterstic polynomial has complex roots. Say $$r=2e^{\pm i\theta}\\\text{where }\theta=\arctan(\sqrt{15})\\\text{for relation }x_n=x_{n-1}-4x_{n-2}$$ Then the solution are given by $$x_n=\alpha2^n\cos{n\theta}+\beta2^n\sin{n\theta}$$ I don't understand why it shouldn't be $$x_n=\alpha2^ne^{in\theta}+\beta2^ne^{-in\theta}$$ or how the two solution are same. Please help.