Question

求证 1+√3*tanχ/(1-√3*tanχ)=cos(χ-π/3)/cos(χ+π/3)

Answer 1

(1+根号3*tanx)/(1-根号3*tanx) (tanx=sinx/cosx)
=(1+根号3*sinx/cosx)/(1-根号3*sinx/cosx) (分子分母同乘以cosx)
=(cosx+根号3*sinx)/(cosx-根号3*x) (分子分母同除以2)
=(1/2*cosx+根号3/2*sinx)/(1/2*cosx-根号3/2*sinx) (cos π/3=1/2,sin π/3=根号3/2))
=(cos(π/3)cosx+sin(π/3)sinx)/(cos(π/3)cosx-sin(π/3)sinx) (分子分母同时用积化和差公式)
=cos(x-π/3)/cos(x+π/3)