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).