Let $n \in \mathbb{N}^*$ and
$$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$
Show that $T _ { n }$ is the unique polynomial with real coefficients satisfying the relation
$$\forall \theta \in \mathbb { R } , \quad T _ { n } ( \cos ( \theta ) ) = \cos ( n \theta ).$$