When simplifying expressions like \( f^{-1}(g(x)) \), first use co-function identities to make \(g\) the same as \(f\). Then, ensure the argument of the inner function (e.g., \( y \) in \( f^{-1}(f(y)) \)) is within the principal value range of \( f^{-1} \). If not, adjust it by adding or subtracting multiples of the function's period (e.g., \(2\pi\) for sin/cos, \(\pi\) for tan).