we have to find all the solutions of $f(x)$ and $g(x)$, given that,
$f(x)\cdot g(x) = f(x) + g(x)$
$(f(x) - 1 )\cdot( g(x) - 1 ) = 1$
I have found out 2 solutions so far, $f(x)= g(x) = 2$, and $f(x) = \sec^2(x)$ , $g(x) = \csc^2(x)$, (and vice versa)