Some Trigonometric Identities Involving Fibonacci and Lucas Numbers

In this paper, using the number of spanning trees in some classes of graphs, we prove the identities:

$\begin{eqnarray*} &&F_n=\frac{2^{n-1}}{n}\sqrt{\prod_{k=1}^{n-1}(1-\cos\frac{k\... ...sin^2{\frac{k\pi}{n}})=L_{2n}-2=F_{2n+2}-F_{2n-2}-2,\;\;n\geq 1, \end{eqnarray*}$

where

and

denote the Fibonacci and Lucas numbers, respectively. Also, we give a new proof for the identity:

$\begin{displaymath}F_n=\prod_{k=1}^{\lfloor\frac{n-1}{2}\rfloor}(1+4\sin^2{\frac... ...or\frac{n-1}{2}\rfloor}(1+4\cos^2{\frac{k\pi}{n}}),\;\;n\geq 4.\end{displaymath}$

Received November 16 2009; revised version received November 26 2009. Published in Journal of Integer Sequences, November 29 2009.

Some Trigonometric Identities Involving Fibonacci and Lucas Numbers

Kh. Bibak and M. H. Shirdareh Haghighi Department of Mathematics Shiraz University Shiraz 71454 Iran

Kh. Bibak and M. H. Shirdareh Haghighi
Department of Mathematics
Shiraz University
Shiraz 71454
Iran