\(\dfrac{a+c}{b+d}=\dfrac{a-c}{b-d}\)
\(\Leftrightarrow ab-ad+cb-cd=ab+ad-cb-cd\)
=>-2ad=-2cb
=>ad=cb
=>a/b=c/d
Đặt a/b=c/d=k
=>a=bk; c=dk
\(\dfrac{a^{2009}-c^{2009}}{b^{2009}-d^{2009}}=\dfrac{b^{2009}k^{2009}-d^{2009}k^{2009}}{b^{2009}-d^{2009}}=k^{2009}\)
\(\left(\dfrac{a}{b}\right)^{2009}=\left(\dfrac{bk}{b}\right)^{2009}=k^{2009}\)
Do đó: \(\dfrac{a^{2009}-c^{2009}}{b^{2009}-d^{2009}}=\left(\dfrac{a}{b}\right)^{2009}\)