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