This identity can be proved directly as follow: \begin{align*} &\cos A + \cos B + \cos C \\ &= (\cos A + \cos B) - \cos (A+B)\\ &= 2\cos\frac{A+B}{2}\cos\frac{A-B}{2} - \big(2\cos^2\frac{A+B}{2} - 1\big)\\ &= 1 + 2\cos\frac{A+B}{2}\big(cos\frac{A-B}{2}-\cos\frac{A+B}{2}\big)\\ &=1 + 2\sin\frac{C}{2}\cdot 2 \sin\frac{A}{2}\sin\frac{B}{2}\\ &= 1 + 4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2} \end{align*}