I have looked at the solution and it simply says that if the number of 'g's and 'f's on both sides of the equation are equal, then the two expressions will be the same given the condition f(x).g(x) = 1.
However I am unable to see how that happens and have been trying to prove this. This is as far as I have come.
Let $f(x) = x$, then $g(x) = 1/x$
$f(f(x)) = x$
$g(g(x)) = x$
$f(g(x)) = \dfrac 1x$
$g(f(x)) = \dfrac 1x$
Then the given expressions become
$g(g(f(f(g(f(x))))) = \dfrac 1x$
$f(f(g(g(f(g(x))))) = \dfrac 1x$
This shows the expressions are equal. But I am stuck now and do not understand how to proceed from here. Trying quadratic expressions do not seem like an option as the expressions get very complex.
Any help regarding how to proceed from here would be greatly appreciated.
Thanks in advance, Bootstrap