Bibak, Kh.; Shirdareh Haghighi, M. H. Some trigonometric identities involving Fibonacci and Lucas numbers. (English) Zbl 1201.11023 J. Integer Seq. 12, No. 8, Article ID 09.8.4, 5 p. (2009). Let \(F_n\) and \(L_n\) be the Fibonacci and Lucas numbers. The authors prove the identities \[ F_n=\frac{2^{n-1}}{n}\sqrt{\prod_{k=1}^{n-1}(1-\cos\tfrac{k\pi}{n}\cos\tfrac{3k\pi}{n})},\quad n\geq 2 \] and \[ \prod_{k=0}^{n-1}(1+4\sin^2\tfrac{k\pi}{n})=L_{2n}-2=F_{2n+2}-F_{2n-2}-2,\quad n\geq 1. \] Reviewer: Florin Nicolae (Berlin) Cited in 3 Documents MSC: 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 05C05 Trees 15A18 Eigenvalues, singular values, and eigenvectors Keywords:Fibonacci numbers; Lucas numbers; trigonometric identity Software:OEIS PDFBibTeX XMLCite \textit{Kh. Bibak} and \textit{M. H. Shirdareh Haghighi}, J. Integer Seq. 12, No. 8, Article ID 09.8.4, 5 p. (2009; Zbl 1201.11023) Full Text: EuDML EMIS Online Encyclopedia of Integer Sequences: Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1. Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. a(n) = Product_{k=1..n} (1 + 4*sin(2*Pi*k/n)^2).