Explore Fourier Series & Jacobi Elliptic Functions: Hints for Sine Expansion

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Hi all. I am currently trying to find the first few terms of a sine expansion of

q(t)= (A+B*cn(t,m))/(C+D*cn(t,m))

where m is the modulus and cn is the jacobic elliptic cnoidal function and A,B,C,D real and C>D implying no poles. I realize that I should start with a simpler problem. Do some of you have any hints for the sinusoidal version:

q(t) = (A+B cos(t))/(C+D cos(t))

How do I handle this sinusoid term in the denominator?

 

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So, I think I prematurely jumped the gun in asking this question. I can do the sinusoidal version. And for the elliptic version, ugh. Yeah. Exactly.