Let $\theta_{kl}$ be an angle such that $\cos\theta_{kl}=\frac{1}{2}(\cos(\frac{2\pi k}{n})+\cos(\frac{2\pi l}{n}))$.
Given that definition, if I introduce a new variable $t$ is the following a correct?
$\cos(t\theta_{kl})\approx\frac{1}{2}(\cos(\frac{2\pi kt}{n})+\cos(\frac{2\pi lt}{n}))$
Update: I'm actually interested in the asymptotics of $\theta_{kl}$. By a second order approximation $\theta_{kl}^2=O(\frac{k^2+l^2}{n^2})$. Is it correct? If it is, then the above holds, right? But $\theta_{kl}$ needs to be small.