Check if two regular expressions are equivalent

Question

I thought I proved that those two regular expressions are and identity using induction. But my friend told me that there was a counterexample. I am not sure anymore.

$$(1(1 + 0)^*)^* = (10^*)^*$$

Answer 1

$1(1 + 0)^*$ is the language of all words over alphabet $\{0,1\}$ which start with $1$. If we split them before the $1$s, we see that $1(1+0)^* = (10^*)^+$. But in general $$(L^+)^* = L^*$$ so you are correct.