If a Language is Non-Recognizable then what about its complement?
Assuming that you can prove statement 1 and 2 above, you have just presented a proof. You're trying to show that there exists a language which is non-recognizable such that its complement is recognizable. You then prove that $!A(TM)$ has this property. Then you want to show that there exists a non-recognizable language whose complement is also non-recognizable and then you show that $EQ(TM)$ has this property.

The point is that to prove that you can't comment, you need only to show that both possibilities are possible ie. that examples of each exist.