tanA＋tanB＋tanC
=sinA/cosA+sinB/cosB+sinC/cosC
=(sinAcosBcosC+sinBcosAcosC+sinCcosAcosB)/cosAcosBcosC
=(cosC(sinAcosB+sinBcosA)+sinCcosAcosB)/cosAcosBcosC
=(cosCsin(A+B)+sinCcosAcosB)/cosAcosBcosC
=(cosCsinC+sinCcosAcosB)/cosAcosBcosC
=sinC(cos(π-A-B)+cosAcosB)/cosAcosBcosC
=sinC(-cos(A+B)+cosAcosB)/cosAcosBcosC
=sinC(-cosAcosB+sinAsinB+cosAcosB)/cosAcosBcosC
=sinCsinAsinB/cosAcosBcosC
=tanAtanBtanC