Symmetric square well, wavefunction is weird

[StarWombat]

Hi,
I'm trying to work my way through some problems and am stuck on one for a symmetric infinite square well, of width 2a, so -a<x<+a. Since this is the symmetric case, the wavefunction should be a linear combination of the terms

(a)^-½ cos (nπx/2a) for odd n,
(a)^-½ sin (nπx/2a) for even n.

The question gives a wavefunction Ψ(x,0) = ε(x) (3a)^-½(1 - cos (2πx/a)) where ε(x) = +1 for x>0, -1 for x<0. I want to rearrange this so it's a linear combination of cos and sin terms, for various values of n. It's easy to get 1-cos(2y) to a single term proportional to sin²(y) using trig identities, but I don't want the square. I want a linear combination of cosine and sine terms where I can read off the value of n for each term (and hence find the probabilities of the various energy eigenvalues). I don't think I can just read n out of the existing cosine because that would imply that n=4, but the cos terms correspond to odd values of n.

And I'm probably sleepy and will kick myself once I see the solution, but at the moment I just can't think of what way to massage this into the linear combination I want. So if anyone has any ideas please shout them out.

Thanks in advance

 

[DrClaude]

You won't be able to do it with trigonometric identities because of the ε(x). Try a Fourier transform.