Let $ P_n $ be the path graph on n vertices named $ \{1, \ldots, n\} $.
Let $ D = diag (1,2,\ldots,2,1) $ and $ A $ be the adjacency matrix of $ P_n $.
Then the Laplacian matrix of $ P_n $ is given by $ L = D - A $, and the normalized Laplacian matrix is given by $ N = D^{-\frac{1}{2}}LD^{-\frac{1}{2}} $.
I want to prove that the eigenvalues of $ N $ are $ 1 - \cos\frac{k\pi}{n-1} $ $ (k = 0, \ldots, n-1) $. How can I prove this?
My attempt:
As an attempt, I was able to prove that the eigenvalues of $ L $ are $ 2(1 - \cos\frac{k\pi}{n}) $ $ (k = 0, \ldots, n-1) $, but I couldn't find the connection between two spectra.