There are only really a few classes of move to consider. Supposing for a second that white plays tenuki, black R2 makes one eye instantly and threatens to make another eye on the next move. If white doesn't play tenuki, you can check yourself that if white plays any one of Q1, R1, R3, S1, S2, T1 or T2, then black R2 is a good response (that makes black live). (Be careful that neither your T3 stones nor your Q2 stones run out of liberties, otherwise you'll have to fill that eye back in. There's some weird-looking stuff with some of the moves above, but ultimately none of it is harmful.)

So, R2 is the move that black really wants, so it's the only interesting move for white (as it's the only one that doesn't fail outright so far!). White R2 is threatening to connect underneath on the next move (e.g. R2 S2 Q1 R3 R1 S1 P1, and black only has a single two-space eye; a throw-in doesn't help, because R2 S2 Q1 R1 S1 R3 Q1 T1 still gives white time to connect out - check that similar things happen if white plays R2 and black responds any of R3, S1, T1 or T2). So we now only have two responses to consider for black: Q1 and R1.

Let's suppose black responds R1. Intuitively, you probably agree that white's interesting moves are the hane (atari) and extending (to S2) - this is more or less because black would like to be able to play S2 on the next move, but these are the two moves that stop that being possible. If R2 R1 S1, then it's unreasonable to allow white to take the stone, so black connects at Q1. Now white can just connect, and the group is dead. (If you're not sure why: later white could atari at P1, black R3, white T2, black captures at T1, and white has a dead shape inside.) If R2 R1 S2, then black S1 allows white to make a dead shape with T2; alternatively, black T2 allows white to make a dead shape with S1. So black R1 - your suggested move - dies.

Let's suppose black responds Q1. If white plays any of R1, R3, S1, T1 or T2, then S2 is a nice eyeshapey move and black lives. So of course white wants to try playing S2. Now if black plays R1, R3, T1 or T2, then white S1 makes a dead shape; so black had better play S1. After this, black T2 would make an eye (check there are no liberty problems), so white plays T2 and prevents it; the best result black can get here (and hence overall) is to connect at R3 for seki (check what happens if black R1 or T1; white should then play T4). Edit: nope - no seki. See Dusk Eagle's post below. Oops!

So - provided white makes no mistakes - black has exactly one way to live in seki, and everything else dies outright. I think.