Sum of Infinite Series of Product of nth Power of cos 2 theta by 2n+1th Multiple of Sine

Theorem
Let $\theta \in \R$ such that $\theta \ne m \pi$ for any $m \in \Z$.

Then:

Proof
Let $\theta \ne \dfrac {m \pi} 2$ for any $m \in \Z$.

Then $\size {\cos 2 \theta} < 1$.