Can you prove this trig identity?

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Discussion Overview

The discussion revolves around proving the trigonometric identity \(\sum_{k=1}^{n-1}\sin\frac{km\pi}{n}\cot\frac{k\pi}{2n} = n-m\) for positive integers \(m\) and \(n\) with \(m \le n\). Participants explore various approaches to the proof, including the application of Euler's formula.

Discussion Character

  • Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant presents the identity to be proven and specifies the conditions for \(m\) and \(n\).
  • Another participant suggests using Euler's formula for sine and cosine as a potential starting point for the proof.
  • A different participant expresses skepticism about the advantages of using Euler's formula, noting difficulties with handling geometric sequences in double summation.
  • There is a question regarding the context of the identity, with one participant inquiring if it is related to schoolwork.
  • Another participant clarifies that the identity is a tool for proving a broader result about sine functions and their coefficients, indicating it is not typical homework material.
  • A participant expresses a desire for more engagement with the proof, questioning the lack of interest in the formula.
  • A link to an external resource is provided, possibly for further reference or context.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of Euler's formula and the context of the identity, indicating that multiple competing approaches and interpretations exist without a clear consensus.

Contextual Notes

Participants have not fully resolved the mathematical steps involved in the proof, and there are varying assumptions about the context in which the identity is applied.

elimqiu
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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]
 
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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??
 
micromass said:
Why not start by applying the Euler formula
What do you get then??

Thanks micromass, not see real advantage yet...geometric sequence cannot be handled easily with double summation...
 
elimqiu said:
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]

What is the context of the question? Is it for schoolwork?
 
berkeman said:
What is the context of the question? Is it for schoolwork?
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 homework in any math course I guess:)
 
No one interested in a proof of such a pretty formula?
 

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