How to verify a trigonometric identities

Question

could you please help me verify this statement,

if, $\xi^k = \cos k\theta + i\sin k\theta$, then,

$\cos k\theta=\dfrac{1}{2}\left(\xi^k+\dfrac{1}{\xi^k} \right)$

$\sin k\theta=-\dfrac{i}{2}\left(\xi^k-\dfrac{1}{\xi^k} \right)$

where $1< k<N$ and $N$ is a real number.

Now, is it correct to say that,

$\cos (k+1)\theta=\dfrac{1}{2}\left(\xi^{(k+1)}+\dfrac{1}{\xi^{(k+1)}} \right)$

$\sin (k+1)\theta=-\dfrac{i}{2}\left(\xi^{(k+1)}-\dfrac{1}{\xi^{(k+1)}} \right)$

Answer 1

Note that

$$\xi^{k} = \cos k\theta + i\sin k\theta$$

$$\xi^{-k} = \cos (-k\theta) + i\sin (-k\theta)=\cos k\theta - i\sin k\theta$$

then

$$\xi^{k}+\xi^{-k}=2 \cos k\theta $$

$$\xi^{k}-\xi^{-k}=2i \sin k\theta $$

and from here identities follow.