Why ask for forgiveness again, right after Tachanun and before the Torah?
Tachanun and Vehu Rachum are long pleas for mercy. Then this short one comes, just before the Torah scroll is taken out. It is fair to ask why it needs to be there.
The older books mostly just list it. Machzor Vitry (a prayer book put together in northern France around 1100) has the leader and the people say it back and forth, and then the ark is opened. Menorat HaMaor (a medieval guide to Jewish practice by Rabbi Yisrael Alnakawa, who died in Toledo, Spain, in 1391) places it right before the Torah comes out on Mondays and Thursdays, and again at the afternoon prayer on a fast day. Neither says why.
The Eliyah Rabbah (a commentary on the Shulchan Aruch, written around 1700) does give a reason. The Torah is about to be read, and we have not lived up to it. God holds back His anger with us anyway, out of great mercy. So we say "God, slow to anger" before the reading.
That is one commentator's explanation, and it was written centuries after the prayer was already in use. It explains the timing. It does not tell us who wrote the words, and none of the sources we opened does.
Machzor Vitry, Order of Shacharit 93,Menorat HaMaor, On Prayer, Tachanun 5 and Fasts 7,Eliyah Rabbah on Shulchan Aruch, Orach Chaim 134:3What would it mean for God to "hide His face" from me? Does that happen?
The words are older than the prayer. "Do not hide Your face from me" is in Tehillim 27:9 and again in 102:3, which adds "in my day of trouble." Abudarham says this line of the prayer is based on it.
In the Torah, hiding the face is a warning. In Devarim 31:17 God says that if the people leave Him for other gods, "I will hide My face from them." They will say: "Is it not because my God is not in my midst that these troubles found me?" Verse 18 says it again, with the verb doubled.
The Ramban (Rabbi Moshe ben Nachman of Spain, who died around 1270) says the second hiding is not the same as the first. In the first, God hides the face of His mercy, and many troubles come. In the second, only the face of redemption is hidden. The people still stand under God's promise, "I will not reject them" (Vayikra 26:44), but the redemption stays hidden until they add full confession and complete repentance to the regret of verse 17.
The Gemara (Chagigah 5b) reads the same verse and finds comfort. Rava, one of its sages, says God is saying: even though I have hidden My face from them, I will speak with them in a dream. Another sage, Rav Yosef, says: His hand is stretched out over us, as Yeshayahu says, "In the shadow of My hand I covered you."
One psalm goes further. Tehillim 22:25 says God "did not hide His face" from the person in pain, "and when he cried out to Him, He heard." The Radak (Rabbi David Kimchi, Provence, died 1235) reads it as what Israel will say after it leaves exile: even in exile, God had not looked down on our suffering.
None of these sources gives a way to tell whether God is hiding or life is just hard. What the prayer does is ask, in so many words, that it not happen.
Tehillim 27:9, 102:3 and 22:25,Devarim 31:17-18,Ramban on Devarim 31:18,Chagigah 5b,Radak on Tehillim 22:25I say "we have sinned" twice a week. Does saying it actually do anything?
The Mishnah describes a public fast (Taanit 2:1). The oldest person there says to the crowd: brothers, the book of Yonah does not say God saw the sackcloth and the fasting of Nineveh. It says God saw "their deeds, that they turned from their evil way." Then he quotes the prophet Yoel (Joel): "Tear your hearts, not your clothes." That is the same verse that says "slow to anger."
The king of Nineveh had ordered everyone to turn from "the violence in their hands" (Yonah 3:8). The Gemara (Taanit 16a) asks what that meant in practice. Shmuel says that even if someone stole a beam and built it into a large building, he must tear down the whole building and give the beam back. That is how Shmuel describes Nineveh. Jewish law ruled that the thief may pay for the beam instead, so that people are not scared away from repenting (Mishnah Gittin 5:5).
Then Rav Adda bar Ahavah, another sage there, gives a picture. Someone who confesses a sin and does not stop is like a person holding an impure crawling creature while he dips in a mikveh (a ritual pool). The creature is a source of impurity, so even all the water in the world would not make him pure. Throw it out of his hand, and one dip in a regular mikveh counts at once.
The Gemara's answer to a question like yours is blunt. The words do not work while the thing is still in your hand. The moment it is out of your hand, they do. The Ashamnu page asks a related question: is it a real confession if I say the same list every day?
Mishnah Taanit 2:1,Yoel 2:13,Yonah 3:8 and 3:10,Taanit 16a,Mishnah Gittin 5:5