All you need to know about Carpenter's Square - 11
How does the extra bend at affect the outcome?
is normally not a good move in standard Carpenter's square shapes covered earlier but it is good now. is important to help form a seki. If plays instead, black simply plays 'a' and white is one liberty too short in the capturing race.
If white plays black needs to be careful. is a calm move. In the end 'a' and 'b' are miai so black is unconditionally alive.
If black is tricked into playing , following and , both forcing moves, is good. Black can win the capturing race by playing 'a' but that is a T-four dead shape. So black corner is completely dead.
from the other side is normally incorrect. is the best point. Apart from , playing anywhere else would result in a big-eye killing shape (please verify). So there is a messy ko fight now. In practice, if black isn't happy with Solution 1 (where all his points are gone), and he has plenty of ko threats, it may be worthwhile to try this diagram. This corner is a two-stage ko, and for black to win an approach move is needed. So winning this ko won't be easy for either side, a good way for black to complicate situations on the board.
Conclusion: Black's corner should be unconditionally alive, unless he chooses other options.