Given $n\in2\Bbb N$, what is precise algebraic relation between $cos\frac{\pi}{n-1}$,$cos\frac{\pi}{n+1}$?
Both numbers are algebraic, which implies there should be an algebraic relation between them. What is this precise relation?
Is this relation $$\frac{T_{n-1}\big(cos\frac{\pi}{n^2-1}\big)}{T_{n+1}\big(cos\frac{\pi}{n^2-1}\big)}=\frac{cos\frac{\pi}{n+1}}{cos\frac{\pi}{n-1}}$$ where $T_n(x)$ is Tchebyshev polynomial of first kind?
These are algebraic numbers, not transcendental, so there are certainly algebraic relations involving them. For example, $T_n(\cos(\pi/n)) = T_m(\cos(\pi/m)) = -1$ where $T_m$ and $T_n$ are Chebyshev polynomials of the first kind.