Let $n \in \mathbb{N}^*$ and
$$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$
For $k \in \llbracket 1 , n \rrbracket$, we set $y _ { k , n } = \cos \left( \frac { ( 2 k - 1 ) \pi } { 2 n } \right)$. Show that
$$T _ { n } ( X ) = 2 ^ { n - 1 } \prod _ { k = 1 } ^ { n } \left( X - y _ { k , n } \right).$$