Question

Cho a, b, c khác nhau đôi một, chứng minh rằng:

a) b-c/(a-b) (a-c) + c-a/(b-c) (b-a) +a-b/(c-a) (c-b) = 2/a-b+2/b-c+2/c-a

Answer 1

\(VT=\dfrac{b-c}{\left(a-b\right)\left(a-c\right)}+\dfrac{c-a}{\left(b-c\right)\left(b-a\right)}+\dfrac{a-b}{\left(c-a\right)\left(c-b\right)}\)

\(VT=\dfrac{b-a+a-c}{\left(a-b\right)\left(a-c\right)}+\dfrac{c-b+b-a}{\left(b-c\right)\left(b-a\right)}+\dfrac{a-c+c-b}{\left(c-a\right)\left(c-b\right)}\)

\(VT=\dfrac{-1}{a-c}+\dfrac{1}{a-b}+\dfrac{-1}{b-a}+\dfrac{1}{b-c}+\dfrac{-1}{c-b}+\dfrac{1}{c-a}\)

\(VT=\dfrac{2}{a-b}+\dfrac{2}{b-c}+\dfrac{2}{c-a}=VP\)