tan(α+β)=(tanα+tanβ)/(1-tanα·tanβ)
1-tanα·tanβ=(tanα+tanβ)/tan(α+β)
tanα·tanβ=1-(tanα+tanβ)/tan(α+β)
左式
=[1-(tan(A/2)+tan(B/2))/tan((A+B)/2)]+[1-(tan(B/2)+tan(C/2))/tan((B+C)/2)]+[1-(tan(C/2)+tan(A/2))/tan((C+A)/2)]
=[1-(tan(A/2)+tan(B/2))/tan((π-C)/2)]+[1-(tan(B/2)+tan(C/2))/tan((π-A)/2)]+[1-(tan(C/2)+tan(A/2))/tan((π-B)/2)]
=[1-(tan(A/2)+tan(B/2))/cot(C/2)]+[1-(tan(B/2)+tan(C/2))/cot(A/2)]+[1-(tan(C/2)+tan(A/2))/cot(B/2)]
=tan(C/2)[cot(C/2)-(tan(A/2)+tan(B/2))]+tan(A/2)[cot(A/2)-(tan(B/2)+tan(C/2))]+tan(B/2)[cot(B/2)-(tan(C/2)+tan(A/2))]
=1-tan(C/2)[tan(A/2)+tan(B/2)]+1-tan(A/2)[tan(B/2)+tan(C/2)]+1-tan(B/2)[tan(C/2)+tan(A/2)]
=3-2[tan(A/2)tan(B/2)+tan(B/2)tan(C/2)+tan(C/2)tan(A/2)]
=3-2·左式
∴左式=1