Find $\sum_{k=0}^6 \cot \left({\pi\over 21} + {k\pi\over 7}\right)$

Question

Find $$\sum_{k=0}^6 \cot \left({\pi\over 21} + {k\pi\over 7}\right)$$

I don't know how to do this at all. I initially started using complex numbers but couldn't get it.

Please give a simple solution.Thanks.

See http://math.stackexchange.com/questions/1607134/cotangent-summation-proof?lq=1 — lab bhattacharjee, Mar 17 '16 at 05:42

score 3 · Accepted Answer · edited Dec 10 '24 at 22:03

3

You can write it as $\dfrac{e^{ix}}{\sin(x)}-i$ . Now we have $\sin(x)=\dfrac{e^{ix}-e^{-ix}}{2i}$. Now can you continue?

edited Dec 10 '24 at 22:03

M. A.

answered Mar 17 '16 at 05:59

Archis Welankar

1 Answers1