Nevermind, I just forgot what cosine is at pi/2...
... Well, even though I solved that last part, I'm now stuck trying to find the value of . I kind of suspect I should be able to get this from Parseval's Theorem, given in Apostol as . I suspect this because, somehow, the teacher claimed to have used this to show that .
Anyway, I plugged in the constant function and got a truism, so in general I can tell I don't want to use constant functions. I suppose I want to find some function such that is equal to the sum I'm looking to evaluate, but I can't see how to make this magically happen.
I'm afraid I don't follow. If this is a proof that I'm not exactly sure what is taken as known, from which we prove this. However, here is my (somewhat failed) attempt at reproducing the professor's argument: Using Parseval, where and and so I get the wrong answer.
In any case, I've used another proof of the same fact, and I'm still not sure how to solve the problem I'm faced with.