# 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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