分式求证题,若a+b=1,求证:a/(b^3-1)-b/(a^3-1)=2(b-a)/a^2b^2+3

Question

分式求证题,
若a+b=1,求证:a/(b^3-1)-b/(a^3-1)=2(b-a)/a^2b^2+3

Answer 1

证：原式左边=(1-a)/(a^3-1)-(1-b)/(b^3-1)
=-1/(a^2+a+1)+1/(b^2+b+1)
=[-(b^2+b+1)+(a^2+a+1)]/[(a^2+a+1)(b^2+b+1)]
=(a^2-b^2+a-b)/[a^2b^2+a^2(b+1)+(a+1)b^2+(a+1)(b+1)]
=[(a+b)(a-b)+a-b]/(a^2b^2+a^2b+a^2+ab^2+b^2+ab+a+b+1)
=2(a-b)/[a^2b^2+ab(a+b)+a^2+b^2+ab+2]
=2(a-b)/(a^2b^2+ab+a^2+b^2+ab+2)
=2(a-b)/[a^2b^2+(a+b)^2+2]
=(2a-2b)/(a^2b^2+3)