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Prove that $$\tan nA=\frac{\binom n1t-\binom n3t^3+\binom n5t^5-\dots}{1-\binom n2t^2+\binom n4t^4-\dots}$$ where $t=\tan A$.

I have a solution of it by de Moivre's theorem but I am looking for a solution by combinatorics. How can I do this. Any help will be appreciated, thanks in advance.

Parcly Taxel
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user449276
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    [Edited to be more realistic] I have just had fun reading http://math.nist.gov/mcsd/Seminars/2008/2008-01-11-Benjamin-presentation.pdf. Towards the end he gives a combinatorial proof that $\cos n\theta=T_n(\cos\theta)$. I do wonder if this one could be attacked in the same sort of way? – ancient mathematician May 25 '17 at 11:10
  • See https://math.stackexchange.com/questions/346368/sum-of-tangent-functions-where-arguments-are-in-specific-arithmetic-series – lab bhattacharjee May 26 '17 at 05:48

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