Bound on the roots of trigonometric equation

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

$$F= \sum a_k\sin^m(x)\cos^n(x)$$ (4.3)

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