Let $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ be the sequence of polynomials defined by $U _ { 0 } = 1 , U _ { 1 } = X - 1$ and $\forall n \in \mathbb { N } , U _ { n + 2 } = ( X - 2 ) U _ { n + 1 } - U _ { n }$. Let $\theta \in \mathbb { R }$. Show that $\forall n \in \mathbb { N } , U _ { n } \left( 4 \cos ^ { 2 } \theta \right) \sin \theta = \sin ( ( 2 n + 1 ) \theta )$.