I was asked to find a solution to $$\frac{\sin^2(nx)}{n^2\sin^2(x)}=2^{-1/2}$$ where $n$ is a fixed integer greater than 1.
Numerically, there's a solution just above 1/n so I decided to find this one (rather than one of the other roots). I hoped to find an obvious pattern that I could use to find some sort of closed-form solution.
It turns out that the power series of the root jumps out, numerically. Since x is only used inside a sine, the power series has only odd terms. The coefficients look like
1.0019063576966069771401037205331682344703361047262779848987578751140847/n +
0.466422280109844953629560577622677335783739044717/n^3 +
0.3247444758314010835749/n^5 + ...
which are annoyingly close to 1, 1/2, 1/3, ... but are clearly distinct.
Any idea how to solve this? This is, I'm sure, well beyond anything the original question-asker cares about, but now my interest is piqued.