<s>[INST] How to prove that $ \prod_{k=1}^{m} (cos(\frac{(2k-1)\pi}{2m+1} )+1) = \frac{2m+1}{2^{m}} $ $$
\mbox{How to prove that
}\quad\prod_{k = 1}^{m}\left\{%
\cos\left(\left[2k - 1\right]\pi \over 2m+1  \right)+1\right\} =
{2m + 1 \over 2^{m}}\quad {\Large ?}
$$
where $m$ is an integer and $1 < m < 90$.
I try to use the identity that
$\cos\left(2\theta\right) + 1  = 2\cos^{2}\left(\theta\right)$, but I'm stuck with
$\prod_{k = 1}^{m}\cos^{2}\left(\left[2k - 1\right]\pi \over 4m + 2\right)$ part.
Please give me an advice, I appreciate it and thank you for every reply.
 [/INST] If $\cos(2m+1)x=0,$
$(2m+1)x=(2n+1)\dfrac\pi2$ where $n$ is any integer
$x=\dfrac{(2n+1)\pi}{2(2m+1)}$ where $0\le n\le2m$
Now using this
$\cos(2m+1)x=2^{2m}\cos^{2m+1}x+\cdots+(-1)^m(2m+1)\cos x$
So, the roots of $$2^{2m}c^{2m+1}+\cdots+(-1)^m(2m+1)c=0$$ are $\cos\dfrac{2(2n+1)\pi}{(2m+1)},0\le n\le2m$
If $\cos x=0,n=m$
So, the roots of $$2^{2m}c^{2m}+\cdots+(-1)^m(2m+1)=0$$ are $\cos\dfrac{(2n+1)\pi}{2(2m+1)},0\le n\le2m, m\ne n$
$\implies2^{2m}\prod_{n=0,n\ne m}^{2m}\cos\dfrac{(2n+1)\pi}{2(2m+1)}=(-1)^m(2m+1)$
As $\pi-\dfrac{(2n+1)\pi}{2(2m+1)}=\dfrac{\pi(1+2(2m-n))}{2(2m+1)}$
$\implies2^{2m}\prod_{n=0}^{m-1}(-1)^m\cos^2\dfrac{(2n+1)\pi}{2(2m+1)}=(-1)^m(2m+1)$
Can you take it from here?
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