I came across this question while doing trigonometry. I have tried everything that I could possibly think of, AM/GM, converting it into quadratic equation, conditional identities, solving from RHS, solving from LHS, however, have not gotten anywhere. Please help me!

Question: In triangle ABC, prove that: $$(\sin A + \sin B)(\sin B+\sin C)(\sin C+\sin A) \gt \sin A \sin B \sin C$$

  • 1
    $\begingroup$ As $0<A<\pi,\sin A>0$ etc. and $$\frac{\sin A+\sin B}2\ge \sqrt{\sin A\sin B}$$ $\endgroup$ – lab bhattacharjee Jul 2 '14 at 16:02
  • $\begingroup$ @labbhattacharjee How exactly did you get that? Where did $\sin c$ go? $\endgroup$ – Gummy bears Jul 2 '14 at 16:05
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    $\begingroup$ @labbhattacharjee: that's only true (aside from the constant) if they are positive. If you already know all of the them are positive, there are easier way (each term on the left is larger than the one on the right). EDIT: opps, had been thinking about $\cos$ the whole time. $\endgroup$ – Gina Jul 2 '14 at 16:05
  • $\begingroup$ @Cookies, Take the product $\endgroup$ – lab bhattacharjee Jul 2 '14 at 16:06
  • $\begingroup$ The product of the left hand side? @labbhattacharjee $\endgroup$ – Gummy bears Jul 2 '14 at 16:08

$\sin A+\sin B>\sin A,\sin B+\sin C>\sin B,\sin C+\sin A>\sin C$. Take product.

  • $\begingroup$ I see now. Don't know how I missed that... I'm horrible at proof :/ $\endgroup$ – Gummy bears Jul 2 '14 at 16:12
  • $\begingroup$ I love this. One line proof. $\endgroup$ – MonK Jul 2 '14 at 17:20

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