$\begin{align}  & \sin A+\sin B+\sin C=2\sin \tfrac{A+B}{2}\cos \tfrac{A-B}{2}+\sin C \\ & \text{Note: }A+B=\pi -C \\ & \sin A+\sin B+\sin C=2\sin \tfrac{\pi -C}{2\grave{\ }}\cos \tfrac{A-B}{2}+\sin C \\ & \sin A+\sin B+\sin C=2\sin (\tfrac{\pi }{2\grave{\ }}-\tfrac{C}{2})\cos \tfrac{A-B}{2}+\sin C \\ & \sin A+\sin B+\sin C=2\cos (\tfrac{C}{2})\cos \tfrac{A-B}{2}+\sin C \\ & \sin A+\sin B+\sin C=2\cos (\tfrac{C}{2})\cos \tfrac{A-B}{2}+2\sin \tfrac{C}{2}\cos \tfrac{C}{2} \\ & \sin A+\sin B+\sin C=2\cos \tfrac{C}{2}\left[ \cos \tfrac{A-B}{2}+\sin \tfrac{C}{2} \right] \\ & \sin A+\sin B+\sin C=2\cos \tfrac{C}{2}\left[ \cos \tfrac{A-B}{2}+\sin \tfrac{\pi -(A+B)}{2} \right] \\ & \sin A+\sin B+\sin C=2\cos \tfrac{C}{2}\left[ \cos \tfrac{A-B}{2}+\sin (\tfrac{\pi }{2}-\tfrac{A+B}{2}) \right] \\ & \sin A+\sin B+\sin C=2\cos \tfrac{C}{2}\left[ \cos \tfrac{A-B}{2}+\cos \tfrac{A+B}{2} \right] \\ & \sin A+\sin B+\sin C=2\cos \tfrac{C}{2}\left[ 2\cos \frac{\tfrac{A-B}{2}+\tfrac{A+B}{2}}{2}\cos \frac{\tfrac{A-B}{2}-\tfrac{A+B}{2}}{2} \right] \\ & \sin A+\sin B+\sin C=2\cos \frac{C}{2}\left[ -2\cos \frac{A}{2}\cos \frac{-B}{2} \right] \\ & \sin A+\sin B+\sin C=4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2} \\\end{align}$