anyone have any clever ideas on how to solve this for x?
Thanks for being so nice about me not explaining in the manner you require, my bad. But as to the 'huh', I was firstly confused at how your statement followed from the equation, and secondly confused at why I would/should observe that fact to solve the equation. Is that the only way? Or do you suggest that to be the easiest way? I don't know. But I do know this follows from that observation -
and since the arccos of both 1 and -1 is (or more specifically and where is even and is odd) I would now have to find - so unless there is an algebraic method for doing that, splitting the equation in two doesn't help me. Is there anything else you can recommend?
implies that for some integer , which then implies that but this implies that:
Similarly (using the other equation): we find that for some integer n:
so every integer determines a solution .
(don't rely on the algebra above being right as I cannot check it in detail at present, either I or another helper will correct any mistakes when we get the chance).