Let $q$ be an arbitrary number from $[0, \pi]$. Consider the summation
$$ S (x) = \frac{1}{L} \sum_{i=0}^L \cos \left(2 \frac{q}{L} k x - \theta(\frac{q}{L} k ) \right) , $$
where the function $\theta$ is defined as
$$ \theta (y) = 2 \arctan \frac{1 }{2 \sin y } . $$
How can I calculate $S $ to the accuracy of $1/L^2$?
The difficulty comes from the $\theta$ function. But it is a slowly varying function. Therefore, one might somehow use the same trick as used in calculating the summation
$$ \frac{1}{L} \sum_{i=0}^L \cos \left(2 \frac{q}{L} k x \right) . $$
The result I guessed is
$$ S(x) \simeq \frac{1}{ 2 L \sin \frac{q}{L} x} \left [\sin(2 q x - \theta(q) + \frac{q}{L} x) - \sin (\frac{q}{L} x) \right] . $$
But I am not sure it is accurate on the order of $1/L^2$. How to make it more systematic?