由正弦定理得a=2RsinA,b=2RsinB,c=2RsinC,故有
2RsinA+2RsinC=a+c=2b=4RsinB,两边约去2R得
sinA+sinC=2sinB=2sin(180-A-C)=2sin(A+C)
即sinA+sinC=2sin(A+C),利用和差化积与倍角公式得
2sin((A+C)/2)cos((A-C)/2)=4sin((A+C)/2)cos((A+C)/2)
即cos((A-C)/2)=2cos((A+C)/2)
再利用和差公式得
cos(A/2)cos(C/2)+sin(A/2)sin(C/2)=2cos(A/2)cos(C/2)-2sin(A/2)sin(C/2)
cos(A/2)cos(C/2)=3sin(A/2)sin(C/2)
两边同除cos(A/2)cos(C/2)得
tanA/2tanC/2=1/3