$\cos A+\cos B+\cos C$
$=2\cos\dfrac{A+B}{2}\cos\dfrac{A-B}{2}+\cos C$
$=2\cos\dfrac{180^o-C}{2}\cos\dfrac{A-B}{2}+\cos 2.\dfrac{C}{2}$
$=2\cos\Big(90^o-\dfrac{C}{2}\Big)\cos\dfrac{A-B}{2}+2\sin\dfrac{C}{2}\cos\dfrac{C}{2}$
$=2\cos\dfrac{C}{2}\Big(\cos\dfrac{A-B}{2}+\sin\dfrac{C}{2}\Big)$
$=2\cos\dfrac{C}{2}.\Big(\cos\dfrac{A-B}{2}+\cos\dfrac{180^o-A-B}{2}\Big)$
$=2\cos\dfrac{C}{2}.2\cos\dfrac{180^o-2A-2B}{4}.\cos\dfrac{-180^o}{4}$
$=2\sqrt2\cos\dfrac{C}{2}.\cos(45^o-\dfrac{A}{2}-\dfrac{B}{2})$
$=2\sqrt2\cos\dfrac{C}{2}.\sin(45^o+90^o+\dfrac{C}{2})$
$=2\sqrt2\cos\dfrac{C}{2}\sin(45^o+\dfrac{C}{2})$
