Show that, for all $x \in \mathbb { R }$ and for all $n \in \mathbb { N } ^ { * }$, $$\left( x ^ { n } - 1 \right) ^ { 2 } = \prod _ { k = 1 } ^ { n } \left( x ^ { 2 } - 2 x \cos \frac { 2 k \pi } { n } + 1 \right)$$
Show that, for all $x \in \mathbb { R }$ and for all $n \in \mathbb { N } ^ { * }$,
$$\left( x ^ { n } - 1 \right) ^ { 2 } = \prod _ { k = 1 } ^ { n } \left( x ^ { 2 } - 2 x \cos \frac { 2 k \pi } { n } + 1 \right)$$