Does a divorce occur when a husband says, "You are divorced" in a state of anger, knowing that the wife is remorseful and repentant and wishes to continue the marital life, and how can she regain her husband's trust and forgiveness?

The divorce has occurred because it is an explicit pronouncement. If the husband had a valid private seclusion (khalwa shar'iyya) with his wife, then according to some scholars, he may take her back during her 'iddah (waiting period). If he had intercourse with her, then he may take her back during her 'iddah. The wife should express her sincere repentance to her husband, for repentance erases what came before it, and he should forgive her.