What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$

Further I wonder if a method can be extended to $$\sum_{k=0}^{n-1}\csc\left(w+ k \frac{\pi}{n}\right)\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$

Here it is shown that,

$$\sum_{k=0}^{n-1}\csc\left(x+\frac{k\pi}n\right)\csc\left(y+\frac{k\pi}n\right)=\frac{n\sin n(x-y)}{\sin(x-y)\sin nx\sin ny}$$

So I thought I could extend this idea using the trigonometry product to sums identity

giving: $$\cos \left(\theta +\frac{\pi k}{n}\right) \cos \left(\frac{\pi k}{n}+\tau \right) \cos \left(\frac{\pi k}{n}+\phi \right)=\frac{1}{4} \left(\cos \left(\theta +\frac{\pi k}{n}+\tau -\phi \right)+\cos \left(\theta +\frac{\pi k}{n}-\tau +\phi \right)+\cos \left(-\theta +\frac{\pi k}{n}+\tau +\phi \right)+\cos \left(\theta +\frac{3 \pi k}{n}+\tau +\phi \right)\right)$$ and $$\cos \left(\theta +\frac{\pi k}{n}\right) \cos \left(\frac{\pi k}{n}+\tau \right) \cos \left(\frac{\pi k}{n}+\omega \right) \cos \left(\frac{\pi k}{n}+\phi \right)=\frac{1}{8} \left(\cos (\theta +\tau -\omega -\phi )+\cos (\theta -\tau -\omega +\phi )+\cos (\theta -\tau +\omega -\phi )+\cos \left(\theta +\frac{2 \pi k}{n}+\tau -\omega +\phi \right)+\cos \left(\theta +\frac{2 \pi k}{n}+\tau +\omega -\phi \right)+\cos \left(\theta +\frac{2 \pi k}{n}-\tau +\omega +\phi \right)+\cos \left(-\theta +\frac{2 \pi k}{n}+\tau +\omega +\phi \right)+\cos \left(\theta +\frac{4 \pi k}{n}+\tau +\omega +\phi \right)\right)$$

and then taking the logarithmic derivative and making use of Chebyshev polynomials however I cannot quite manipulate this to work.

What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(w+ k \frac{\pi}{n}\right)\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$

Further I wonder if a method can be extended to $$\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$

Here it is shown that,

$$\sum_{k=0}^{n-1}\csc\left(x+\frac{k\pi}n\right)\csc\left(y+\frac{k\pi}n\right)=\frac{n\sin n(x-y)}{\sin(x-y)\sin nx\sin ny}$$

So I thought I could extend this idea using the trigonometry product to sums identity

giving: $$\cos \left(\theta +\frac{\pi k}{n}\right) \cos \left(\frac{\pi k}{n}+\tau \right) \cos \left(\frac{\pi k}{n}+\phi \right)=\frac{1}{4} \left(\cos \left(\theta +\frac{\pi k}{n}+\tau -\phi \right)+\cos \left(\theta +\frac{\pi k}{n}-\tau +\phi \right)+\cos \left(-\theta +\frac{\pi k}{n}+\tau +\phi \right)+\cos \left(\theta +\frac{3 \pi k}{n}+\tau +\phi \right)\right)$$ and $$\cos \left(\theta +\frac{\pi k}{n}\right) \cos \left(\frac{\pi k}{n}+\tau \right) \cos \left(\frac{\pi k}{n}+\omega \right) \cos \left(\frac{\pi k}{n}+\phi \right)=\frac{1}{8} \left(\cos (\theta +\tau -\omega -\phi )+\cos (\theta -\tau -\omega +\phi )+\cos (\theta -\tau +\omega -\phi )+\cos \left(\theta +\frac{2 \pi k}{n}+\tau -\omega +\phi \right)+\cos \left(\theta +\frac{2 \pi k}{n}+\tau +\omega -\phi \right)+\cos \left(\theta +\frac{2 \pi k}{n}-\tau +\omega +\phi \right)+\cos \left(-\theta +\frac{2 \pi k}{n}+\tau +\omega +\phi \right)+\cos \left(\theta +\frac{4 \pi k}{n}+\tau +\omega +\phi \right)\right)$$

and then taking the logarithmic derivative and making use of Chebyshev polynomials however I cannot quite manipulate this to work.

What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(w+ k \frac{\pi}{n}\right)\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$

Further I wonder if a method can be extended to $$\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$

Here it is shown that,

$$\sum_{k=0}^{n-1}\csc\left(x+\frac{k\pi}n\right)\csc\left(y+\frac{k\pi}n\right)=\frac{n\sin n(x-y)}{\sin(x-y)\sin nx\sin ny}$$

So I thought I could extend this idea using the trigonometry product to sums identity

giving: $$\cos \left(\theta +\frac{\pi k}{n}\right) \cos \left(\frac{\pi k}{n}+\tau \right) \cos \left(\frac{\pi k}{n}+\phi \right)=\frac{1}{4} \left(\cos \left(\theta +\frac{\pi k}{n}+\tau -\phi \right)+\cos \left(\theta +\frac{\pi k}{n}-\tau +\phi \right)+\cos \left(-\theta +\frac{\pi k}{n}+\tau +\phi \right)+\cos \left(\theta +\frac{3 \pi k}{n}+\tau +\phi \right)\right)$$ and $$\cos \left(\theta +\frac{\pi k}{n}\right) \cos \left(\frac{\pi k}{n}+\tau \right) \cos \left(\frac{\pi k}{n}+\omega \right) \cos \left(\frac{\pi k}{n}+\phi \right)=\frac{1}{8} \left(\cos (\theta +\tau -\omega -\phi )+\cos (\theta -\tau -\omega +\phi )+\cos (\theta -\tau +\omega -\phi )+\cos \left(\theta +\frac{2 \pi k}{n}+\tau -\omega +\phi \right)+\cos \left(\theta +\frac{2 \pi k}{n}+\tau +\omega -\phi \right)+\cos \left(\theta +\frac{2 \pi k}{n}-\tau +\omega +\phi \right)+\cos \left(-\theta +\frac{2 \pi k}{n}+\tau +\omega +\phi \right)+\cos \left(\theta +\frac{4 \pi k}{n}+\tau +\omega +\phi \right)\right)$$

and then taking the logarithmic derivative and making use of Chebyshev polynomials however I cannot quite manipulate this to work.

What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$

Further I wonder if a method can be extended to $$\sum_{k=0}^{n-1}\csc\left(w+ k \frac{\pi}{n}\right)\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$

Here it is shown that,

$$\sum_{k=0}^{n-1}\csc\left(x+\frac{k\pi}n\right)\csc\left(y+\frac{k\pi}n\right)=\frac{n\sin n(x-y)}{\sin(x-y)\sin nx\sin ny}$$

So I thought I could extend this idea using the trigonometry product to sums identity

giving: $$\cos \left(\theta +\frac{\pi k}{n}\right) \cos \left(\frac{\pi k}{n}+\tau \right) \cos \left(\frac{\pi k}{n}+\phi \right)=\frac{1}{4} \left(\cos \left(\theta +\frac{\pi k}{n}+\tau -\phi \right)+\cos \left(\theta +\frac{\pi k}{n}-\tau +\phi \right)+\cos \left(-\theta +\frac{\pi k}{n}+\tau +\phi \right)+\cos \left(\theta +\frac{3 \pi k}{n}+\tau +\phi \right)\right)$$ and $$\cos \left(\theta +\frac{\pi k}{n}\right) \cos \left(\frac{\pi k}{n}+\tau \right) \cos \left(\frac{\pi k}{n}+\omega \right) \cos \left(\frac{\pi k}{n}+\phi \right)=\frac{1}{8} \left(\cos (\theta +\tau -\omega -\phi )+\cos (\theta -\tau -\omega +\phi )+\cos (\theta -\tau +\omega -\phi )+\cos \left(\theta +\frac{2 \pi k}{n}+\tau -\omega +\phi \right)+\cos \left(\theta +\frac{2 \pi k}{n}+\tau +\omega -\phi \right)+\cos \left(\theta +\frac{2 \pi k}{n}-\tau +\omega +\phi \right)+\cos \left(-\theta +\frac{2 \pi k}{n}+\tau +\omega +\phi \right)+\cos \left(\theta +\frac{4 \pi k}{n}+\tau +\omega +\phi \right)\right)$$

and then taking the logarithmic derivative and making use of Chebyshev polynomials however I cannot quite manipulate this to work.

What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$

Further I wonder if a method can be extended to $$\sum_{k=0}^{n-1}\csc\left(w+ k \frac{\pi}{n}\right)\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$

Here it is shown that,

$$\sum_{k=0}^{n-1}\csc\left(x+\frac{k\pi}n\right)\csc\left(y+\frac{k\pi}n\right)=\frac{n\sin n(x-y)}{\sin(x-y)\sin nx\sin ny}$$

So I thought I could extend this idea using the trigonometry product to sums identity

giving: $$\cos \left(\theta +\frac{\pi k}{n}\right) \cos \left(\frac{\pi k}{n}+\tau \right) \cos \left(\frac{\pi k}{n}+\phi \right)=\frac{1}{4} \left(\cos \left(\theta +\frac{\pi k}{n}+\tau -\phi \right)+\cos \left(\theta +\frac{\pi k}{n}-\tau +\phi \right)+\cos \left(-\theta +\frac{\pi k}{n}+\tau +\phi \right)+\cos \left(\theta +\frac{3 \pi k}{n}+\tau +\phi \right)\right)$$ and $$\cos \left(\theta +\frac{\pi k}{n}\right) \cos \left(\frac{\pi k}{n}+\tau \right) \cos \left(\frac{\pi k}{n}+\omega \right) \cos \left(\frac{\pi k}{n}+\phi \right)=\frac{1}{8} \left(\cos (\theta +\tau -\omega -\phi )+\cos (\theta -\tau -\omega +\phi )+\cos (\theta -\tau +\omega -\phi )+\cos \left(\theta +\frac{2 \pi k}{n}+\tau -\omega +\phi \right)+\cos \left(\theta +\frac{2 \pi k}{n}+\tau +\omega -\phi \right)+\cos \left(\theta +\frac{2 \pi k}{n}-\tau +\omega +\phi \right)+\cos \left(-\theta +\frac{2 \pi k}{n}+\tau +\omega +\phi \right)+\cos \left(\theta +\frac{4 \pi k}{n}+\tau +\omega +\phi \right)\right)$$

and then taking the logarithmic derivative and making use of Chebyshev polynomials however I cannot quite manipulate this to work.

What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(w+ k \frac{\pi}{n}\right)\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$

Further I wonder if a method can be extended to $$\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$

Here it is shown that,

$$\sum_{k=0}^{n-1}\csc\left(x+\frac{k\pi}n\right)\csc\left(y+\frac{k\pi}n\right)=\frac{n\sin n(x-y)}{\sin(x-y)\sin nx\sin ny}$$

So I thought I could extend this idea using the trigonometry product to sums identity

giving: $$\cos \left(\theta +\frac{\pi k}{n}\right) \cos \left(\frac{\pi k}{n}+\tau \right) \cos \left(\frac{\pi k}{n}+\phi \right)=\frac{1}{4} \left(\cos \left(\theta +\frac{\pi k}{n}+\tau -\phi \right)+\cos \left(\theta +\frac{\pi k}{n}-\tau +\phi \right)+\cos \left(-\theta +\frac{\pi k}{n}+\tau +\phi \right)+\cos \left(\theta +\frac{3 \pi k}{n}+\tau +\phi \right)\right)$$ and $$\cos \left(\theta +\frac{\pi k}{n}\right) \cos \left(\frac{\pi k}{n}+\tau \right) \cos \left(\frac{\pi k}{n}+\omega \right) \cos \left(\frac{\pi k}{n}+\phi \right)=\frac{1}{8} \left(\cos (\theta +\tau -\omega -\phi )+\cos (\theta -\tau -\omega +\phi )+\cos (\theta -\tau +\omega -\phi )+\cos \left(\theta +\frac{2 \pi k}{n}+\tau -\omega +\phi \right)+\cos \left(\theta +\frac{2 \pi k}{n}+\tau +\omega -\phi \right)+\cos \left(\theta +\frac{2 \pi k}{n}-\tau +\omega +\phi \right)+\cos \left(-\theta +\frac{2 \pi k}{n}+\tau +\omega +\phi \right)+\cos \left(\theta +\frac{4 \pi k}{n}+\tau +\omega +\phi \right)\right)$$

and then taking the logarithmic derivative and making use of Chebyshev polynomials however I cannot quite manipulate this to work.