The Chebyshev polynomials of the first kind $(T_n)_{n \in \mathbb{N}}$ are defined by $T_n(\cos\theta) = \cos(n\theta)$. By noting that for every real $\theta$, we have $e^{in\theta} = \left(e^{i\theta}\right)^n$, show that $$\forall n \in \mathbb{N}, \quad T_n = \sum_{0 \leqslant k \leqslant n/2} \binom{n}{2k} \left(X^2 - 1\right)^k X^{n-2k}$$ Write in Maple or Mathematica language a function $T$ taking as argument a natural integer $n$ and returning the expanded expression of the polynomial $T_n$.
The Chebyshev polynomials of the first kind $(T_n)_{n \in \mathbb{N}}$ are defined by $T_n(\cos\theta) = \cos(n\theta)$.
By noting that for every real $\theta$, we have $e^{in\theta} = \left(e^{i\theta}\right)^n$, show that
$$\forall n \in \mathbb{N}, \quad T_n = \sum_{0 \leqslant k \leqslant n/2} \binom{n}{2k} \left(X^2 - 1\right)^k X^{n-2k}$$
Write in Maple or Mathematica language a function $T$ taking as argument a natural integer $n$ and returning the expanded expression of the polynomial $T_n$.