Is the following relation concerning fibonacci numbers, $F_n$ true? $$F_{2n-1}^n=2^{2n^2}\prod\limits_{r=1}^{n}\prod\limits_{s=1}^{n}\left(\cos^2\frac{r\pi}{2n+1}+\cos^2\frac{s\pi}{2n+1}\right)$$
I am dumbfounded seeing this expression. Is the expression true. If so, should we try to prove the expression taking only the portions inside the brackets, or should we make use of de-moivre's theorem, or, any recurrence formulae? Meanwhile, I know of this relation among Fibonacci numbers and trigonometric function: $$F_n=\prod\limits_{k=1}^{\lfloor\frac{n-1}{2}\rfloor}\left(1+4\cos^2\frac{k\pi}{n}\right)$$. How could we use this expression in proving the above relation. The main relation arose from the formulae for finding the number of tilings of a chessboard using dominoes. Any hints? Thanks beforehand.
Too long to answer the comment: This is not true. The right hand side counts the number of domino tilings of a square with dimensions $2n\times 2n$ and the left hand side can be thought as stacking $n$ times a stripe of dimensions $2\times 2n.$ In general there is no way you can go from one to another because there are configurations that can not be seen as a stripe of dominos. For example:
Notice also that the left hand side for $n=3$ is $6728$ and the left hand side is $25$ they are coprime, so you can not multiply anything to fix it. In particular consider the following tiling.
Check that there is no way you can form this by just multiplying Fibonacci numbers.