I've come across this expression as part of a very separate problem I've been working on: $$\frac{\cosh(4n\theta)-1}{\sinh((2n+1)\theta)+\sinh((2n-1)\theta)}$$ and noticed that whenever one sets $\theta: \sinh(\theta)\in \Bbb N$, the expression always comes out an integer. This leads me to believe there is a nice(-ish?) simplification for this, but I've had no luck finding it thus far.
I know the denominator can be rewritten as $2\sinh(2n\theta)\cosh(\theta)$ but I couldn't see this getting anywhere.