The Chebyshev polynomials of the second kind $(U_n)_{n \in \mathbb{N}}$ are defined by $U_n = \frac{1}{n+1} T_{n+1}'$, and satisfy $U_n(\cos\theta) = \frac{\sin((n+1)\theta)}{\sin\theta}$ for $\theta \notin \pi\mathbb{Z}$. Deduce the following properties: a) The sequence $(U_n)_{n \in \mathbb{N}}$ satisfies the same recurrence relation $T_{n+2} = 2X T_{n+1} - T_n$ as the sequence $(T_n)_{n \in \mathbb{N}}$. b) For every natural integer $n$, the polynomial $U_n$ is split over $\mathbb{R}$ with simple roots belonging to $]-1,1[$. Determine the roots of $U_n$.
The Chebyshev polynomials of the second kind $(U_n)_{n \in \mathbb{N}}$ are defined by $U_n = \frac{1}{n+1} T_{n+1}'$, and satisfy $U_n(\cos\theta) = \frac{\sin((n+1)\theta)}{\sin\theta}$ for $\theta \notin \pi\mathbb{Z}$.
Deduce the following properties:
a) The sequence $(U_n)_{n \in \mathbb{N}}$ satisfies the same recurrence relation $T_{n+2} = 2X T_{n+1} - T_n$ as the sequence $(T_n)_{n \in \mathbb{N}}$.
b) For every natural integer $n$, the polynomial $U_n$ is split over $\mathbb{R}$ with simple roots belonging to $]-1,1[$. Determine the roots of $U_n$.