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A trig identity [tough]

  1. Sep 9, 2011 #1
    Show that [itex]\displaystyle{\sum_{k=1}^{n-1}\sin\frac{km\pi}{n}\cot\frac{k\pi}{2n} = n-m}\quad\quad(m,n\in\mathbb{N}^+,\ m\le n)[/itex]
     
    Last edited: Sep 9, 2011
  2. jcsd
  3. Sep 9, 2011 #2

    micromass

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    Why not start by applying the Euler formula

    [tex]\cos \alpha= \frac{e^{i\alpha}+e^{-i\alpha}}{2},~\sin \alpha=\frac{e^{i\alpha}-e^{-i\alpha}}{2i}[/tex]

    What do you get then??
     
  4. Sep 9, 2011 #3
    Thanks micromass, not see real advantage yet...geometric sequence cannot be handled easily with double summation...
     
  5. Sep 9, 2011 #4

    berkeman

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    What is the context of the question? Is it for schoolwork?
     
  6. Sep 9, 2011 #5
    It's a tool to prove

    [itex]f(x)=a_1\sin x+\cdots+a_n\sin nx,\quad |f(x)|\le |\sin x|\quad (\forall x\in\mathbb{R})\implies |a_1+\cdots+a_n|\le 1[/itex]

    It's not fit for hw in any math course I guess:)
     
  7. Sep 10, 2011 #6
    No one interested in a proof of such a pretty formula?
     
  8. Sep 12, 2011 #7
    Last edited by a moderator: Apr 26, 2017
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