200 Followers Problem: Trigonometric sum from the future

Geometry Level 5

$\large \text{S} = \sum_{k=1}^{2020} \left(\dfrac{1}{\sin \left(\frac{2k-1}{4040}\pi + \theta \right) \cdot \sin \left(\frac{2k-1}{4040} \pi - \theta \right)}\right)$

where $$0< \theta < \dfrac{\pi}{2}$$

If $$\text{S}$$ can be represented as:

$\large A \cdot \left( \dfrac{\tan (B \theta)}{\sin (C \theta)} \right)$

where $$A$$, $$B$$ and $$C$$ are positive integers, evaluate $$A+B+C$$.

Original

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