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Prove the given identity.

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

I though of many ideas but could not start even.I know there is an answer for this here. I saw the answer but it is inefficient and out of my mind. A shorter answer is expected. Thanks.

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  • Although the ques is exact duplicate, see what I have stated. –  Jan 07 '17 at 07:40
  • @dxiv I believe OP'S asking for a simpler answer. – Yiyuan Lee Jan 07 '17 at 07:41
  • @Yiyuan Lee, You got me correct. I am asking for a simpler and possibly short answer. –  Jan 07 '17 at 07:42
  • That's what even the answerer of that question also requested for. –  Jan 07 '17 at 07:43
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    @EiNsTeIn If you believe the question must have a simpler, more "efficient" answer then please state your reasons why you believe so. Just because you don't like the answer to the other question doesn't make yours any less of a duplicate. – dxiv Jan 07 '17 at 07:43
  • @dxiv, just see there. Even the poster of that ques. has asked some more hints even after getting the answer, and moreover the answerer had also remarked that there would be some other better solutions. –  Jan 07 '17 at 07:47
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    Then the "approved" methods for generating such a solution are essentially to A) generate one yourself, or B) post a bounty on the question to incentivize someone capable of doing so. I understand that you're probably not in a position to do either at the moment, but as @dxiv says, generally posting a duplicate question isn't the solution (plus: What if your question gets a solution? Will you notify everyone on the other question who was interested, including the four people who have marked it a Favorite question of theirs?) – pjs36 Jan 07 '17 at 07:52
  • @pjs36, I did not get this: (plus:........?) Would u please make it clear? –  Jan 07 '17 at 07:56
  • It has been pointed out that there is interest in a more elegant solution from people participating in the other thread. My hypothetical situation is this: Suppose that this thread gets the elegant solution. Will you take it upon yourself to let those who are interested due to the other thread know that a more elegant solution has been found? How would the OP of that question know there is an answer here, or the person who posted the other answer, or the multiple people who have marked that question a Favorite of theirs? – pjs36 Jan 07 '17 at 08:05
  • I will definetly try to notify them by some means. But the primary requirement is that first this thread should get an efficient answer. –  Jan 07 '17 at 08:08
  • Instead of marking the question as duplicate, one should also consider the intention of OP. Though the same question was asked by me, but I wasn't satisfied with the answer that I got. – pi-π Jan 07 '17 at 12:05

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