Example 2 Given that $f(x)$ is an even function defined on $(-1,1)$, and it is increasing on $(0,1)$. If $f(a-2)-$ $f\left(4-a^{2}\right)<0$, try to determine the range of real number $a$.

Analyzing the transformation of functional inequalities into ordinary algebraic inequalities essentially involves the application of the odd-even properties and monotonicity of functions.
From $f(a-2)-f\left(4-a^{2}\right)0\end{array}\right.$, we solve to get $\sqrt{3}<a<2$ or $2<a<\sqrt{5}$.