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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