Prove Triangle Inequality: $\sum_{cyc} \sin A$

[lfdahl]

Prove, that for any triangle:\[ \sum_{cyc}\sin A - \prod_{cyc}\sin A \ge \sum_{cyc}\sin^3 A \]

 

[lfdahl]

I am so sorry, that I have posted a challenge, the solution of which, I am not certain. My problem is the use of the Rearrangement Inequality in the proof. I have asked in the forum: ”Pre-University Math/trigonometry” (http://mathhelpboards.com/trigonometry-12/usage-rearrangement-inequality-trigonometric-expression-20998.html#post95174), and I am aware, that it is bad policy to post a challenge in one forum and ask for a partial solution of it in another on the MHB site. Again, I am very sorry about this. It won´t happen again.

Here is the suggested solution:

Spoiler

Division by $\sin A \sin B \sin C$:

\[\sum_{cyc}\frac{1}{\sin B \sin C}-1 \geq \sum_{cyc}\frac{\sin^2 A}{\sin B \sin C}=\sum_{cyc}\frac{1- \cos^2 A}{\sin B \sin C} \\\\ \Rightarrow \sum_{cyc}\frac{\cos^2 A}{\sin B \sin C} \geq 1\]

Now, here comes the moment, where the Rearrangement Inequality is applied:

\[\sum_{cyc} \frac{\cos^2A}{\sin B\sin C}\geq \sum_{cyc} \frac{\cos B \cos C}{\sin B\sin C} =\sum_{cyc}\cot B \cot C = 1.\]