Can you prove this trig identity?

[elimqiu]

Show that $\displaystyle{\sum_{k=1}^{n-1}\sin\frac{km\pi}{n}\cot\frac{k\pi}{2n} = n-m}\quad\quad(m,n\in\mathbb{N}^+,\ m\le n)$

 

[micromass]

Why not start by applying the Euler formula

$$\cos \alpha= \frac{e^{i\alpha}+e^{-i\alpha}}{2},~\sin \alpha=\frac{e^{i\alpha}-e^{-i\alpha}}{2i}$$

What do you get then??

 

[elimqiu]

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...

 

[berkeman]

Show that $\displaystyle{\sum_{k=1}^{n-1}\sin\frac{km\pi}{n}\cot\frac{k\pi}{2n} = n-m}\quad\quad(m,n\in\mathbb{N}^+,\ m\le n)$

What is the context of the question? Is it for schoolwork?

 

[elimqiu]

What is the context of the question? Is it for schoolwork?

It's a tool to prove

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

It's not fit for homework in any math course I guess:)

 

[elimqiu]

No one interested in a proof of such a pretty formula?

 

[elimqiu]

http://elinkage.net/math/viewtopic.php?f=5&t=519"