I came across the following equality of regular expressions $$(r|s)^* = (r^*.s^*)^*\,,$$ where $r$ and $s$ are two regular expressions and $.$ denotes concatenation, but I don't know how could I prove it.
My try:
I call $L(r)$ the language defined by the regular expression $r$. And I want to use the fact that two regular expressions $r$ and $s$ are equal iff $L(r) = L(s)$.
The LHS can be writen as follows: $$ L((r|s)^{*}) = \bigcup_{i=0}^{+\infty} L(r|s)^{i} = \bigcup_{i=0}^{+\infty} (L(r)^{i} \cup L(s)^{i})\,. $$
The RHS can be writen as follows: $$ L((r^{*}.s^{*})^{*}) = \bigcup_{i=0}^{+\infty} L((r^{*}.s^{*}))^{i}\,.$$
But I got stuck at this point and I don't know how to proceed.