I'm trying to create a model which is of the form
$$y = (a_0 + a_1l)[b_0+\sum_{m=1}^M b_m\cos(mx-\alpha_m)] [c_0 +\sum_{n=1}^N c_n\cos(nz-\beta_n)]$$
In the above system, $l$,$x$ and $z$ are independent variables and $y$ is the dependent variable. The $a$, $b$ and $c$ terms are the unknowns. To solve for these unknowns, I have two separate data sets that I can use. Using data set $1$ creates an overdetermined system providing me with more observations than unknowns, while data set $2$ creates an underdetermined system with less observations than unknowns. In such a case, which approach would be better - underdetermined or overdetermined? and Why?
An important question is how did you get the sets? Are they observations of some physical phemomena? And why are they distinct sets as opposed to one large set if they are observations?
In general an overdetermined system is no problem assuming the results are consistent you just get some equations which end up being superfluous. The issue you might have though is that the observations are not internally consistent. This might be either because your model isn't quite right or just due to measurement error. In that case I'm assuming there are some number theoretic methods of dealing with such problems using some fitting procedures but someone with more knowledge of numerical methods might be better to answer that.
I will also point out that the problem as you state it is very much not linear. Both due to the cosines and the fact that you're taking a product of the variables (after applying the cosines but still).