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Can anyone help me find a formal reference for the following identity about the summation of squared tangent function:

$$ \sum_{k=1}^m\tan^2\frac{k\pi}{2m+1} = 2m^2+m,\quad m\in\mathbb{N}^+. $$

I have proved it, however, the proof is too long to be included in a paper. So I just want to refer to some books or published articles.

I also found it to be a special case of the following identity,

$$ \sum_{k=1}^{\lfloor\frac{n-1}{2}\rfloor}\tan^2\frac{k\pi}{n} = \frac16(n-1)(-(-1)^n (n + 1) + 2 n - 1),\quad n\in\mathbb{N}^+ $$

which is provided by Wolfram.

Thank you very much!

  • 2
    I can't give you a reference but check this question out. http://math.stackexchange.com/questions/2339/prove-that-sum-k-1n-1-tan2-frack-pi2n-fracn-12n-13?rq=12012-10-20
  • 0
    Thanks Aleks, it is pretty good to find this proof. However I think I'd better find a reference for it.2012-10-20
  • 0
    It's similar to Proof 9 in [Robin Chapman's list of ways to evaluate $\zeta(2)$](http://empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf).2013-07-31

2 Answers 2