求证tanA+B/2tanC/2的值为常数表达式

Question

求证tanA+B/2tanC/2 的值为 常数表达式

Answer 1

tan(π/2-α)=cotα
(tan(π/2-α)=sin(π/2-α)/cos(π/2-α)=(-cosa)/(-sina)=cota
∵在ΔABC中,A+B+C=π 即 A+B=π-C (A+B)/2=(π-C)/2=π/2-C/2
∴tan(A+B)/2=tan(π/2-C/2)=cotC/2
∴tan(A+B)/2*tanC/2
=cotC/2*tanC/2
=1
得证