Number of roots of trigonometric equation

The purpose of this section is to present an algorithm which enable to determine how many roots has an equation $F$ in the unknown $x$ of the form:

$$F= \sum a_k\sin^m(x)\cos^n(x)$$ (4.1)

with $m$ in $[0,M]$ and $n$ in $[0,N]$, both $m, n$ being integers, and $x$ in $(\underline{x},\overline{x})$.