A Menon-type Identity derived using Cohen-Ramanujan sum

Menon’s identity is a classical identity involving gcd sums and the Euler totient function $\phi$. We derived the Menon-type identity ${\sum }_{\begin{array}{c}m=1\\ (m\mathrm{.}{n}^s{)}_s=1\end{array}}^{n^s}(m-1,n^s{)}_s={\mathrm{\Phi }}_s(n^s){\tau }_s(n^s)$ in Czechoslovak Math. J., 72(1):165-176 (2022) where ${\mathrm{\Phi }}_s$ denotes the Klee’s function and $(a,b{)}_s$ denotes a a generalization of the gcd function. Here we give an alternate method to derive this identity using the concept of Cohen-Ramanujan sum.