Let $A_n\in\mathbb{R}^{n\times n}$ be defined as $$ A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 & b & a & \cdots & \cdots & 0 & 0\\ \vdots & \vdots & \vdots &\ddots & \ddots & \vdots & \vdots \\ \vdots & \vdots & \vdots &\ddots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 &\cdots & \cdots & a & b \\ 0 & 0 & 0 &\cdots & \cdots & b & a\end{bmatrix}, $$ where $a,b\in\mathbb{R}$. It is well-known that the eigenvalues of $A_n$ are $$ \text{eig}(A_n) =\left\{ a+2b\cos\left(\frac{\pi}{n+1}k\right), \ k=1,2,\dots,n \right\}. $$

My question.Does there exist a closed-form expression for the eigenvalue density of the sequence $\{A_n\}$ as $n\to \infty$?