Stuck when tackling the computation of $\Phi_n(\zeta_8)$ My current way of calculation of $\Phi_n(\zeta_8)$ where $\Phi_n(x)$ is the $n$-th cyclotomic polynomial and $\zeta_8=\cos(\frac{2\pi}{8})+i\sin(\frac{2\pi}{8})$ leave me now stuck at the problem of calculating $$\prod_{d|n}\sin(\frac{d2\pi}{16})^{\mu(\frac{n}{d})}$$ the product running over all integer divisors 0f $n$. $\mu(n)$ is the Möbius function.
My tests with WolframAlpha Online resulted for e.g. prime $n$ and $n$ squarefree with maximally $2$ prime divisors the for me astonishing results of $\pm 1$,$\pm \cot(\frac{\pi}{8})$,$\pm \cot^2(\frac{\pi}{8})$,$\pm \tan(\frac{\pi}{8})$,$\pm \tan^2(\frac{\pi}{8})$ so far.
I see no way how I could find a closed form for the product which the CAS calculations lead me to conjecture to exist.   

I think I derived a proof of the following formula for $n>1$. I did also a CAS test run for $n=1$ to $n=262$ confirming the formula. $$\Phi_n(\zeta_8)=\zeta_{16}^{\phi(n)}\prod_{d|n}\sin(d\frac{2\pi}{16})^{\mu(\frac{n}{d})}$$
