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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

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  • $\begingroup$ @Andrei, I know that's the solution. But I did not get that. Its too long and confusing... $\endgroup$ – Ger Wyn Oct 15 '16 at 17:29
  • $\begingroup$ It is a consequence of the (cosine addition)=(sine product) formulas and a particular Gauss sum. $\endgroup$ – Jack D'Aurizio Oct 15 '16 at 18:53
  • $\begingroup$ @Jack D'Aurizio, I did not understand. What do you want to say? $\endgroup$ – Ger Wyn Oct 16 '16 at 10:36
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use the following identity $$\sin x={\frac {e^{ix}-e^{-ix}}{2i}}$$

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

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