Let $G$ be a graph with $n$ vertices. Denote by $L(G)$ the Laplacian matrix of $G$ and $0=\lambda_1\leqslant\lambda_2\leqslant...\leqslant\lambda_n$ its spectrum. The number $\lambda_2$ is called the algebraic connectivity of $G$ and is denoted by $a(G)$.

In my research I need the value of $a(P_n)$, where $P_n$ is the path with $n$ vertices. I know (and I have a proof) that $a(P_n)=4\sin^2(\pi/(2n))$. It seems that this result is very well-known and classical, but I cannot find a proof in the literature (in some papers the authors just write $a(P_n)=4\sin^2(\pi/(2n))$ without proof).

Thus I would be very grateful for the reference to a proof of the result (if the proof exists in the literature).