Why do I insert 2015 anywhere I like? #1

Geometry Level 5

\[\large{\sec \left( \dfrac{2\pi}{2015} \right) + \sec \left( \dfrac{4\pi}{2015} \right) + \sec\left( \dfrac{6\pi}{2015} \right) + \ldots + \sec \left( \dfrac{2014\pi}{2015} \right) = \ ? }\]


Bonus : Generalize the sum \(\displaystyle \sum_{k=1}^n \sec \left( \dfrac{2k\pi}{2n+1} \right) \) in terms of \(n\) with a proper derivation without using methods like Induction.

Try it's sister problem: Why do I insert 2015 anywhere I like? #2
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