# Simplifying the Language of this DFA

Above's the DFA in question (Sipser, Page 36). I have obtained the language of this DFA to be 0*1(1+00+01)*. But Sipser's textbook goes on to explain that the language of this DFA is (0+1)*1(00)*. But I just cannot derive Sipser's language from what I have obtained. Any help at all would be appreciated.

The automaton is deterministic, so any string over $$\{0,1\}$$ has a unique path.
We need at least one $$1$$ to move from $$q_1$$ to the component where is the accepting state.
Immediately after reading $$1$$ we are always in accepting state $$q_2$$.
Look at the last $$1$$ in the input. At that moment we accept in $$q_2$$. To return to the accepting state, only reading $$0$$'s, we need an even number of $$0$$'s.