[tex]4sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}=2(cos\frac{A-B}{2}-cos\frac{A+B}{2})sin\frac{C}{2}[/tex]
=[tex]2cos\frac{A-B}{2}sin\frac{C}{2}-2sin^{2}\frac{C}{2}[/tex]
=[tex]sin\frac{A+C-B}{2}+sin\frac{C-A+B}{2}-1+cosC[/tex]
=[tex]sin(\frac{\pi }{2}-B)+sin(\frac{\pi }{2}-A)-1+cosC[/tex]
=cosA+cosB+cosC-1