Is there any closed form of
$$\prod\_{k=1}^{n}\left(\cos(kx)-1\right)?$$
I failed to find references on this problem in the internet.



Closed form of $\prod_{k=1}^{n}\left(\cos(kx)-1\right)$
$$\prod\_{k=1}^{n}\left(\cos kx-1\right)=2^{-n} e^{-\frac{1}{2} i n (n+1) x} \left(e^{i x};e^{i x}\right)\_n^2,$$
with $(\cdot;\cdot)\_n$ the [q-Pochhammer symbol.](https://en.wikipedia.org/wiki/Q-Pochhammer_symbol)  

I guess this counts as a "closed form", but of course it's just a rewriting of the product in terms of some named quantity.




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Steven Stadnicki has suggested to compute the Fourier coefficients,
$$\prod\_{k=1}^{n}\left(\cos kx-1\right)=\tfrac{1}{2}a\_{n,0}+\sum\_{p=1}^{n(n+1)/2}a\_{n,p}\cos px.$$
I find
$$a\_{n,p}=2^{1-n}T\_{n,q},\;\;q=\tfrac{1}{2}n(n+1)+p,$$
with $T\_{n,q}$ the coefficients in [OEIS:A304080](https://oeis.org/A304080).


 Mathematica code to test this:



```
a[n_,p_]:=2^(1-n)*CoefficientList[Expand[Product[(1-x^j)^2,{j,1,n}]],x][[n*(n+1)/2+p+1]]

Product[Cos[k*x]-1,{k,1,n}]-a[n,0]/2-Sum[a[n,p]*Cos[p*x],{p,1,n*(n+1)/2}]//FullSimplify    

```


