已知A+B+C=k180,k∈Z,且tanA,tanB,tanC都有意义,求证：tanA+tanB+tanC=tanAtanBtanC

Question

已知A+B+C=k 180,k∈Z,且tanA,tanB,tanC都有意义,求证：tanA+tanB+tanC=tanA tanB tanC

Answer 1

根据和角的正切公式 tan(a±b)=(tana±tanb)/(1-tana*tanb),
得tanA=tan(k*180-(C+B))=(tan(k*180)-tan(C+B))/(1-tan(k*180)*tan(C+B)=-tan(C+B) (因tan(k*180)=0)
tanA+tanB+tanC=-tan(C+B)+tanB+tanC=(tanB+tanC)/(tanB*tanC-1)+tanB+tanC=(tanB+tanC)*tanB*tanC/(tanB*tanC-1)
tanA•tanB•tanC=(tanB+tanC)/(tanB*tanC-1)•tanB•tanC=上式
故原式得证