KöMaL 2005 B3856

Show that if $a^2$+$b^2$=$a^2 b^2$, and |$a$|$\ne$1, |$b$|$\ne$1, then $\frac{a^7}{{(1-a)}^2}- \frac{a^7}{{(1+a)}^2}= \frac{b^7}{{(1-b)}^2}- \frac{b^7}{{(1+b)}^2}.$