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 \(\sum \limits _{\begin{array}{c} m=1\\ (m,n^s)_s=1 \end{array}}^{n^s} (m-1,n^s)_s=\Phi _s(n^s)\tau _s(n^s)\) in [Czechoslovak Math. J., 72(1):165-176 (2022)] where \(\Phi _s\) 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 properties of the Cohen-Ramanujan sum defined by E. Cohen.