I have the following transcendental equation which may very well lack an analytic solution. I would, at the least, like an expression for the relationship between $\theta$ and $\phi$ in some closed form:
$\cos\left((n+\frac{1}{2})\theta\right)\cosh\left((n-\frac{1}{2})\phi\right) = \cos\left((n-\frac{1}{2})\theta\right)\cosh\left((n+\frac{1}{2})\phi\right)$
Both $\cos((x-1/2)\theta)$ and $\cosh((x-1/2)\phi)$ are eigenvectors defined on the interval $x\in (1,2,...,n)$.
It might help that it seems that the derivatives with respect to x match at $x=n$, but I currently lack proof.
Any special functions that could be used to re-express this or a known method of solution would be useful.