# Counting roots of oscillatory functions

Consider the functions $$f_m : [0,1] \to \mathbb{R}$$ defined by

$$f_m(x) = \displaystyle\sum_{n=1}^{m} \frac{\sin(nx)}{1+\sin^2(n)}$$

for every natural number $$m$$. My task is to find the number of roots of this function for every $$m$$. After some thought, I think it behaves similarly to the function $$g_m(x) = \sum_{n=1}^{m}\sin(nx) = \frac{\sin{(m+1)x/2}}{\sin{(x/2)}}\sin{(mx/2)}$$ which has roots $$x= 2k\pi/m$$, $$0 \le k \le m/2\pi$$ and $$x = 2l\pi/(m+1), 0 \le l \le (m+1)/2k\pi$$. This gives $$\left\lfloor \frac{m}{2\pi}\right\rfloor+\left\lfloor \frac{m+1}{2\pi}\right \rfloor+1$$ roots in the interval $$[0,1]$$ for the function $$g_m$$.

This did match the number of roots of $$f_m$$ for some values of $$m$$, however, inconsistencies were noticed for values of $$m$$ near $$31$$. Roots of $$g_m$$ tell us that the number of roots must keep increasing as $$m$$ increases, which doesn't seem to be true for $$f_m$$. The number of roots decreased from $$9$$ to $$8$$ from $$m=30$$ to $$31$$, and $$11$$ to $$9$$ from $$m=32$$ to $$33$$. Any hints on how to proceed with this problem? We perhaps require an appropriate correction term with $$g_m$$.

$$\sin(nx) = \sin(x) U_{n-1}(\cos(x))$$ where $$U_k$$ are the Chebyshev polynomials of the second kind. So (besides $$0$$) you're asking for the number of roots of the polynomial $$P_m(t) = \sum_{n=1}^m \frac{U_{n-1}(t)}{1+\sin^2(1) U_{n-1}(\cos(1))^2}$$ in the interval $$[0, \cos(1)]$$.
But because of the involvement of the transcendental constant $$\cos(1)$$, I doubt that you'll get a "closed form" for the number of roots.
This is not an answer but it may be helpful. There is a known formula for the number of roots of a function $$f$$ in the interval $$[a,b]$$, namely:
$$N(a,b) = \int_a^b \delta (f(x) ) \vert f'(x) \vert dx \, ,$$
where $$\delta$$ is Dirac's delta. As it is I'm not sure that this formula is easier to deal with than explicitly counting the roots (with some numerical algorithm). For example you can thing of approximating the Dirac delta with a peaked Gaussian and estimate the integral.