Trigo!

Geometry Level 4

If \(A\), \(B\) and \(C\) are positive such that \(A+B+C = \pi\). Find for which value of \(A\), \(B\) and \(C\) that minimizes the following.

\[\large \frac{\cos \left(\frac{A-B}{2}\right)}{\cos \left(\frac{A+B}{2}\right)} + \frac{\cos \left( \frac{B-C}{2}\right)}{\cos \left(\frac{B+C}{2}\right)} + \frac{\cos \left(\frac{C-A}{2}\right)}{\cos \left(\frac{C+A}{2}\right)}\]

If the answer is of the form \(\dfrac{a \pi}{b}\), find \(a+b\), where \(a\) and \(b\) are coprime positive integers.

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