Trigo!

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.