The series \sum a_n\cos(n\theta) + \sum b_n\sin(n\theta) and \sum a_n\sin(n\theta + \pi/2) + \sum b_n\sin(n\theta) are literally, exactly the same; it is purely

The series \sum a_n\cos(n\theta) + \sum b_n\sin(n\theta) and \sum a_n\sin(n\theta + \pi/2) + \sum b_n\sin(n\theta) are literally, exactly the same; it is purely a matter of convention which one you choose to write. In more programming-type language, 'cosine' is just syntactic sugar (or 'sine' is, if you prefer; or, I would rather say, both are, for the real and complex parts of the exponential); but sometimes syntactic sugar is handy!