Problem

The key is to use the constraint that $A + B + C = \pi$ to convert the sum of cosines into a product form. Start by rewriting $\cos A + \cos B$ using sum-to-product formulas, then substitute $C = \pi - (A+B)$ to simplify $\cos C$. This transforms the left side into an expression involving sines and cosines of half-angles, which naturally connect to the inradius $r$ and circumradius $R$ through the standard triangle relations $r = 4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$ and similar identities.

The key is to use the constraint that $A + B + C = \pi$ to convert the sum of cosines into a product form. Start by rewriting $\cos A + \cos B$ using sum-to-product formulas, then substitute $C = \pi - (A+B)$ to simplify $\cos C$. This transforms the left side into an expression involving sines and cosines of half-angles, which naturally connect to the inradius $r$ and circumradius $R$ through the standard triangle relations $r = 4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$ and similar identities.