A solution of the Mathieu equation is of the form $y_1(z) = e^{uz} \phi(z,\sigma)$
$\phi(z,\sigma)$ is to be given by -
$\phi(z,\sigma) = sin(z-\sigma) + s_3sin(3z-\sigma) + s_5sin(5z-\sigma) + .... + c_3cos(3z-\sigma) + c_5cos(5z-\sigma) + ....$
where $\sigma$ is dependent on $a,q$ as are $c,s$.
The above equation therefore must have all periodic terms in the R.H.S. Why would the inclusion of $\cos(z - \sigma)$ introduce a non-periodic term?
Edit:
For reference, I have attached a screenshot of the text.
P.S: I could probably use some better tags for this question. Please edit the question with the appropriate tags if possible or leave a comment below so that I can change it myself.