I suggest to rely on a recursive solution like that in the attached spreadsheet. It is based on the Vincenty's formula which provides accurate avaluation (1/10^-6) of the distance between two points of given lat and long (dec degrees).
instructions:
Allow macro to run.
provide lat and lon in cells a3 and b3 respectively.
provide the step (dec degrees)
Push the green button.
read area (m^2) with the desired approximation .
there are many sites in which you can get the Vincenty's algorithm to adapt it to your needs. My solution is but an example.
Very simple: it's just the angular part of the volume element in spherical polar coordinates exchanging latitude for polar angle: if a is longitude and b is latitude, dOmega = cos(b) db da.
You can also apply the Gauss-Bonnet formula to the "spherical rectangle". The "parallels"(latitudes) are not geodesics, but they have constant geodesic curvature (a simple trigonometric function depending on the latitude)