## $S=\left \{z\in\mathbb{C}| (z+i)^{n}=(z-i)^{n} \right \}$ $S=?$

If $$n\in\mathbb{N}, n\geq2$$ and $$S=\left \{z\in\mathbb{C}| (z+i)^{n}=(z-i)^{n} \right \}$$ then $$S=?$$ The right answer is $$S=\left \{ctg\frac{k\pi}{n} |1\leq k\leq n-1;k\in\mathbb{N}\right \}$$

I started like this $$(\frac{z+i}{z-i})^{n}=1$$ How to continue ?Some ideas?