If $m$ and $n$ be positive integers Prove that $\varphi(mn)= \varphi(m,n) \cdot \varphi[m,n]$ where [m, n] =l.c.m of $a$ and $b$ And (m, n)=g.c.d of $a$ and $b$

My approach $\varphi (mn)=\varphi ((m, n) [m, n])$ $\implies ({mn/[m, n]} ,[m, n])$ $=({mn/[m, n]},{mn/(m, n)}$ Am I in right direction