# A trig identity [tough]

1. Sep 9, 2011

### elimqiu

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

Last edited: Sep 9, 2011
2. Sep 9, 2011

### 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??

3. Sep 9, 2011

### elimqiu

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

4. Sep 9, 2011

### Staff: Mentor

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

5. Sep 9, 2011

### elimqiu

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 hw in any math course I guess:)

6. Sep 10, 2011

### elimqiu

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

7. Sep 12, 2011

### elimqiu

Last edited by a moderator: Apr 26, 2017