Solution (1) Black’s task in this problem is to prevent 2 and a from becoming two eyes for White. Since the eye at a is already secure, this means Black needs to prevent the eye at 2; for this, the throw in of 1 is the key move. If White responds with 2, black 3 kills.
Solution (2) White, however, can resist a bit with the block of 2 here. Black is forced to capture with 3, and White can then further cockroach his way out with 4–10. Finally, only the tail of the white group can be cleanly captured.
In the Solution (2) diagram, instead of 9 Black can arguably try to pincer White on the upper side, trying to prevent it from escaping; whether this works will then depend on what else there is on the board on the upper side.
Solution added.
In the Solution (2) diagram, instead of 9 Black can arguably try to pincer White on the upper side, trying to prevent it from escaping; whether this works will then depend on what else there is on the board on the upper side.