So there seems to be some disconnect between the text book and my professor's way of doing things.

I'm studying for the exam and going over the homework solutions he has posted, and it looks to me like he is using some tricks to find the solutions rather than performing the procedures outlined in the text.
I can't seem to figure out how he arrived at most of his solutions. I need some guidance here.

Here are some of the past homework questions and his solutions:

1]
Fourier Analysis-homeworkq1.jpg
SOLUTION:
Fourier Analysis-homeworka1.jpg

So here I see that he has used Euler's identity to turn A*sin(pi*t) into the form A*(ej*pi*t-e-j*pi*t)/(2*j) and from that found a1 and a-1 by inspecting the coefficients of each exponential function respectively, but I don't understand why he found those since the question pertains to y(t). Furthermore, what does ak=0 mean? I understand that a=0 because there is no DC component, but I'm not sure what he's getting at there.
Second, how did he arrive at the values for b? It appears just by inspection he was able to see that they are related to one another.
And last I thought power was given by Parseval's theorem, but it looks like he just squares the magnitude of the coefficients? And I realize they will be the same value but why did he choose a-1 to do this operation and not a1?

According to our text the equations to be used for these operations are
Fourier Analysis-anformula.jpg
Fourier Analysis-a0formula.jpg
Fourier Analysis-bnformula.jpg
I have more questions but I thought it would be best if I figured this out first, then hopefully I'll be able to answer my other questions myself. Thanks!