Given $A+B+C=4S.$

Now $\cos \left(2S-A\right)+\cos \left(2S-B\right)-\cos \left(2S-C\right)-\cos 2S$

$=\left[\cos \left(2S-A\right)+\cos \left(2S-B\right)\right]-\left[\cos \left(2S-C\right)+\cos 2S\right]$

Using $\cos \theta +\cos \varphi =2\cos \left(\frac{\theta +\varphi }{2}\right)\cos \left(\frac{\theta -\varphi }{2}\right),$

$=2\cos \left(\frac{2S-A+2S-B}{2}\right)\cos \left(\frac{2S-A-2S+B}{2}\right)-2\cos \left(\frac{2S-C+2S}{2}\right)\cos \left(\frac{2S-C-2S}{2}\right)$

$=2\cos \left(\frac{4S-A-B}{2}\right)\cos \left(\frac{B-A}{2}\right)-2\cos \left(\frac{4S-C}{2}\right)\cos \left(\frac{-C}{2}\right)$

Using the given value and $\cos \left(-\theta \right)=\cos \theta ,$

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

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

Using $\cos \theta -\cos \varphi =2\sin \left(\frac{\theta +\varphi }{2}\right)\sin \left(\frac{\varphi -\theta }{2}\right),$

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

$=2\cos \left(\frac{C}{2}\right)\left[2\sin \left(\frac{{\displaystyle B}{\displaystyle -}{\displaystyle A}+{\displaystyle A}{\displaystyle +}{\displaystyle B}}{4}\right)\sin \left(\frac{{\displaystyle A}{\displaystyle +}{\displaystyle B}-{\displaystyle B}{\displaystyle +}{\displaystyle A}}{4}\right)\right]$

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

$=4\sin \left(\frac{A}{2}\right)\sin \left(\frac{B}{2}\right)\cos \left(\frac{C}{2}\right).$