Show that the Fourier series S(x) of:
The answer to your original quaetion is because:
The first term on the right is even and so has a Fourier series with cosine terms only and the second is already in Fourier form and indeed is the right most term in the given series.
(there may be errors in the LaTeX above, but I can't read the rendered images on this machine, but they look OK on CodeCogs)
All I have done is split your function into the sum of an odd and even fuction. It is convienient that the odd part is already in the form of a Fourier series and so needs no further work.