Trig sum

Algebra Level pending

If a geometric progression has \(a_{1}=2.tg (\frac{\pi}{5})\) as its first term and \(q=tg^{2} (\frac{\pi}{5})\) as its ratio of progression, the sum of all its terms, for \(n\) terms, with \(n\) varying from \(1\) to \(\infty\), can be expressed as \(\zeta=tg (\frac{s.\pi}{r})\), such that \(s\) and \(r\) are coprime positive integers. So, what's \(s+r\)?

Details and assumptions

You might desconsider answers out of the interval \([0,\frac{\pi}{2}]\)

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