grandes-ecoles 2021 Q2

grandes-ecoles · France · centrale-maths2__official Proof Proof by Induction or Recursive Construction
Show that, for all $n \in \mathbb{N}$ and $\theta \in \mathbb{R}$, $T_n(\cos\theta) = \cos(n\theta)$.
The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$. 
Show that, for all $n \in \mathbb{N}$ and $\theta \in \mathbb{R}$, $T_n(\cos\theta) = \cos(n\theta)$.

The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.
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