Suppose $z = \cos θ + i \sin θ$. If $n$ is an integer, evaluate $z^n + \bar{z}^n$ and $z^n − \bar{z}^n $.

Question

Suppose $z = \cos θ + i \sin θ$. If $n$ is an integer, evaluate $z^n + \bar{z}^n$ and $z^n − \bar{z}^n $.

My attempt:

Let $z\in \mathbb{C}$ such that $z=\cos\theta +i\sin\theta$ then $\bar{z}=\cos\theta - i\sin\theta$

Then, using Mouvre Form, we have:

$z^n+\bar{z}^n=(\cos\theta +i\sin\theta)^n+(\cos\theta -i\sin\theta)^n=(\cos n\theta+i\sin n\theta)+(\cos n\theta-i\sin n\theta)=2\cos n\theta.$

Analogous:

$ z^n-\bar{z}^n=(\cos\theta +i\sin\theta)^n-(\cos\theta -i\sin\theta)^n=(\cos n\theta+i\sin n\theta)-(\cos n\theta-i\sin n\theta)=2i\sin n\theta$

Is this correct?