Al-Fatihah is waived for the person praying behind the imam in two cases
Question

If I enter the mosque when the imam is bowing and I bow with him, does that rak’ah count for me, even though I have not recited al-Fatihah? If I join him before he bows but then he says takbeer and bows when I have not completed al-Fatihah, what should I do? Should I bow with him and not complete al-Fatihah, or should I complete al-Fatihah and then bow?.
Praise be to Allah.In the answer to question no. 10995 we have stated that reciting al-Fatihah is a pillar or essential part of the prayer for everyone who prays, whether his is praying as an imam, behind an imam, or on his own, both in prayers when Quran is recited out loud and those in which it is recited silently. 
The evidence for that is the report narrated by al-Bukhaari (756) from ‘Ubaadah ibn al-Saamit (may Allah be pleased with him), that the Messenger of Allah (peace and blessings of Allah be upon him) said: “There is no prayer for the one who does not recite the Opening of the Book.” 
See al-Majmoo’ 3/283-285 
The obligation to recite al-Fatihah is not waived for the one who is praying behind the imam except in two cases: 
1 – If he catches up with the prayer when the imam is bowing, in which case he should bow with him, and that rak’ah counts for him even if he did not recite al-Fatihah. 
The evidence for that is the hadeeth of Abu Bakrah (may Allah be pleased with him) who said that he came to the Prophet (peace and blessings of Allah be upon him) when he was bowing and he bowed before he reached the row. He mentioned that to the Prophet (peace and blessings of Allah be upon him) and he said: “May Allah make you more keen, but do not do it again.” Narrated by al-Bukhaari, 783. 
The point here is that if catching up with bowing did not count as catching up with the rak’ah with the imam, the Prophet (peace and blessings of Allah be upon him) would have told him to make up that rak’ah in which he did not catch up with the recitation, but no such thing has been reported from him. This indicates that the one who catches up with bowing has caught up with the rak’ah. 
See Silsilat al-Ahaadeeth al-Saheehah, p. 230. 
The second case in which the obligation to recite al-Fatihah is waived from the one who is praying behind an imam is: 
If he joins the imam in prayer just before he bows, and he is not able to complete al-Fatihah. In that case he should bow with him and not complete al-Fatihah, and that rak’ah counts for him. 
Al-Shiraazi (may Allah have mercy on him) said in al-Muhadhdhab: If he catches up with him when he is standing but he fears that he will miss out on reciting (al-Fatihah), he should leave the opening du’aa’ and focus on reciting al-Fatihah, because that is obligatory and he should not be distracted from it by something that is supererogatory. If he recites part of al-Fatihah then the imam bows, there are two scholarly points of view: one is that he should bow and stop reciting, because following the imam is more important; hence if he catches up with him when he is bowing, the obligation to recite al-Fatihah is waived. The second view is that he should complete al-Fatihah because he started reciting it and he should finish it. End quote. Al-Majmoo’, 4/109 
Shaykh Ibn Baz (may Allah have mercy on him) was asked: If I join the prayer just before the imam bows, should I start reciting al-Fatihah or recite the opening du’aa’? If the imam bows before I complete al-Fatihah, what should I do? 
He replied: 
Reciting the opening du’aa’ is Sunnah and reciting al-Fatihah is obligatory for the one who is praying behind the imam, according to the correct scholarly opinion. If you fear that you will miss al-Fatihah, then start with it, and if the imam bows before you have completed it, then bow with him, and the rest of it is waived for you, because the Prophet (peace and blessings of Allah be upon him) said: “The imam has been appointed to be followed, so do not differ from him. When he says takbeer then say takbeer; when he bows then bow.” Agreed upon. End quote. 
Majmoo’ Fatawa Ibn Baz, 11/243-244 
The Standing Committee was asked: If a worshipper catches up with the congregation and the imam is reciting Quran after al-Fatihah in a prayer where he recites out loud, such as Maghrib, should he recite al-Fatihah or not? If he catches up with the imam when he is standing, and he recites “Al-hamdu Lillaahi Rabb il-‘Aalameen (Praise be to Allah, the Lord of the Worlds)” only, then the imam says takbeer, should he bow with him or complete his recitation? 
They replied: 
Reciting al-Fatihah in prayer is obligatory for the imam, the one who is praying alone and the one who is praying behind an imam, in prayers where Quran is recited silently and those where it is recited out loud, because of the general meaning of the evidence about reciting al-Fatihah in the prayer. If a person comes to the congregation and says takbeer with the imam, then he must recite it. If the imam bows before he has completed it, he has to follow the imam, and that rak’ah counts for him, as is the case when a person catches up with the imam when he is bowing – the rak’ah in which he caught up with the imam bowing counts for him, according to the more correct of the two scholarly opinions, and the obligation to recite al-Fatihah is waived for him, because he is not able to recite it. That is based on the well known hadeeth of Abu Bakrah which is narrated in Saheeh al-Bukhaari. End quote. 
Fatawa al-Lajnah al-Daa’imah, 6/387 
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked about a person who joins the prayer after the imam has finished the opening takbeer and recitation of al-Fatihah, then he starts to recite al-Fatihah but then the imam bows. Should he bow too, or complete his recitation of al-Fatihah? 
He replied: 
If a person joins the prayer and the imam wants to bow, and the person praying behind him has not completed his recitation of al-Fatihah, if only one verse or so is left, and he can complete it and join the imam in bowing, that is fine. If he has a lot left and reciting it means that he will not catch up with the imam in bowing, then he should bow with the imam even if he does not complete al-Fatihah. End quote. 
Majmoo’ Fatawa Ibn ‘Uthaymeen, 15/106 
See also al-Sharh al-Mumti’, 3/242-248.