In $\Delta ABC,\cos A+\cos B+\cos C=$

$\cos A+\cos B+\cos C=2\cos \left(\frac{{\displaystyle A+B}}{{\displaystyle 2}}\right)\cos \left(\frac{{\displaystyle A-B}}{{\displaystyle 2}}\right)+1-2{\sin }^2\frac{C}{2}$

$=1+2\sin \frac{C}{2}\left[-\sin \frac{C}{2}+\cos \left(\frac{A-B}{2}\right)\right]$

$=1+2\sin \frac{C}{2}\left[-\cos \left(\frac{A+B}{2}\right)+\cos \left(\frac{A-B}{2}\right)\right]$

$=1+4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

$\begin{array}{l}=1+\frac{4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}}{R}\\ =1+\frac{r}{R}\end{array}$