The function
is continuous and strictly monotonically increasing in every interval [-pi/2,pi/2]+k*pi while the function
g(x) = (a-x.^2) ./ (b + (a-x.^2)*c)
is continuous and strictly monotonically decreasing everywhere except at its unique pole xp.
Because f and g have opposite monotonicity in each interval [-pi/2,pi/2]+k*pi not containing xp, and because f(x) ranges from -inf to +inf there, the equation
will have exactly one solution in each of those intervals. These solutions are identically the roots of f(x)-g(x). The anomalous interval [-pi/2,pi/2]+k0*pi containing the pole xp can be handled by splitting it into 2 subintervals to the left and right of xp. Each of those subintervals must have exactly one root by the same monotonicity arguments.
Thus, you need simply loop over the first successive n of these intervals,subdividing the k0-th interval appropriately. In each interval, use fzero to find the unique root there.
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