Trigonometric identity between sines of multiples of $\pi/70$

Question

Prove that:

$$\left(\sin\frac {9\pi}{70}+ \sin\frac {29\pi}{70} - \sin\frac {31\pi}{70}\right) \left(\sin\frac {\pi}{70}-\sin\frac {11\pi}{70} - \sin\frac {19\pi}{70}\right) =\frac {\sqrt {5} -4}{4}$$

I Could not get any idea to solve. Please help

@Andrei, I know that's the solution. But I did not get that. Its too long and confusing... — Ger Wyn, Oct 15 '16 at 17:29
It is a consequence of the (cosine addition)=(sine product) formulas and a particular Gauss sum. — Jack D'Aurizio, Oct 15 '16 at 18:53
@Jack D'Aurizio, I did not understand. What do you want to say? — Ger Wyn, Oct 16 '16 at 10:36

score 0 · Answer 1 · answered Oct 15 '16 at 17:14

0

use the following identity $$\sin x={\frac {e^{ix}-e^{-ix}}{2i}}$$

answered Oct 15 '16 at 17:14

E.H.E

23,280

I could not get that. Could you please elaborate a bit more.? – Ger Wyn Oct 15 '16 at 17:32

Trigonometric identity between sines of multiples of $\pi/70$

1 Answers1