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\begin{abstract}
In this paper, using the number of spanning trees in  some
classes of graphs, we prove the identities:
\begin{eqnarray*}
  &&F_n=\frac{2^{n-1}}{n}\sqrt{\prod_{k=1}^{n-1}(1-\cos\frac{k\pi}{n}\cos \frac{3k\pi}{n}}),\;\;n\geq 2,\\
  &&\prod_{k=0}^{n-1}(1+4\sin^2{\frac{k\pi}{n}})=L_{2n}-2=F_{2n+2}-F_{2n-2}-2,\;\;n\geq 1,
\end{eqnarray*}
where $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers,
respectively. Also, we  give a new proof for the identity:
$$F_n=\prod_{k=1}^{\lfloor\frac{n-1}{2}\rfloor}(1+4\sin^2{\frac{k\pi}{n}})=\prod_{k=1}^{\lfloor\frac{n-1}{2}\rfloor}(1+4\cos^2{\frac{k\pi}{n}}),\;\;n\geq 4.$$
\end{abstract}

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