Question

Cho a/b =c/d chứng minh a*b/c*d = a^2-b^2/c^2-z^2

Answer 1

\(\dfrac{a}{b}=\dfrac{c}{d}\\ \Rightarrow\dfrac{a}{c}=\dfrac{b}{d}\\ \Rightarrow\dfrac{a^2}{c^2}=\dfrac{b^2}{d^2}\)

Áp dụng t/c của DTSBN , ta có :

\(\dfrac{a^2}{c^2}=\dfrac{b^2}{d^2}=\dfrac{a^2-b^2}{c^2-d^2}\\ \Rightarrow\dfrac{a^2}{c^2}=\dfrac{a^2-b^2}{c^2-d^2}\)

mà \(\dfrac{a^2}{c^2}=\dfrac{a}{c}\cdot\dfrac{a}{c}=\dfrac{a}{c}\cdot\dfrac{b}{d}=\dfrac{ab}{cd}\) (Do `a/c=b/d`)

nên \(\dfrac{ab}{cd}=\dfrac{a^2-b^2}{c^2-d^2}\)