For $p \in \mathbb{N}^*$, we set $$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$ We denote by $P_p$ the polynomial such that, for all real $x$, $P_p(x) = \det(x I_p - A_p)$. Let $x \in \mathbb{R}$ such that $|2 - x| < 2$. After justifying the existence of a unique $\theta \in ]0, \pi[$ such that $2 - x = 2\cos\theta$, determine $P_p(x)$ as a function of $\sin((p+1)\theta)$ and $\sin(\theta)$.
For $p \in \mathbb{N}^*$, we set
$$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$
We denote by $P_p$ the polynomial such that, for all real $x$, $P_p(x) = \det(x I_p - A_p)$.
Let $x \in \mathbb{R}$ such that $|2 - x| < 2$. After justifying the existence of a unique $\theta \in ]0, \pi[$ such that $2 - x = 2\cos\theta$, determine $P_p(x)$ as a function of $\sin((p+1)\theta)$ and $\sin(\theta)$.