I came across this in an idea I was working and was looking for any ideas on an exact evaluation or equivalences.

The blue and the green function are the same, the only differences are one is in terms of sin and the other csc, and one the index is run from 1 while the other 2.

I'm looking for when the index starts at either 1 or 2, and ends at a number, say j. A formula in terms of j and x would be best, however, if you can see an answer where j is infinity, and thus the sum is just an expression of x, that's fine also.

I messed around with it for a bit and got nowhere. Wolfram says the ratio test is inconclusive. Anyway, thought I'd see if anyone had any ideas or knowledge about it's form.