Solutions to an inequality involving trigonometric terms

Question

I would like to determine the angles $\theta\in [0,2\pi[$ such that $$ \big(1-\cos(n\theta)\big)\big(1-\cos(m\theta)\big)\geq 1/4 $$ for every positive integers $m,n\in \mathbb N\setminus \{ 0\}$.

Answer 1

There aren't any, even if we restrict the condition to $m=n$.

If $\theta$ is not a rational multiple of $2\pi$, then by choosing suitable $m$ we can make $\cos(m\theta)$ arbitrarily close to $1$ (or anything else between $-1$ and $1$).

So we can confine attention to $\theta$ a rational multiple of $2\pi$. Thus we may assume that $\theta=2\pi\frac{a}{b}$ for integers $a$ and $b$, with $b$ positive.

Let $m=b$. Then $\cos(m\theta)=1$.