The takbeer for starting the prayer may count for the takbeer for bowing too

Praise be to Allah.

If a person catches up with the imam when he is bowing, his first takbeer will be sufficient, according to the more correct of the two scholarly opinions, because it is the more important obligation, and the greatest pillar or essential part of the prayer, without which the prayer is not valid. With regard to the takbeer for bowing, it was said that it is obligatory, and it was said that it is Sunnah, so it is of less importance that the first takbeer. If there is no time, the first takbeer, which is a essential part of the prayer, will suffice for the takbeer of bowing too. But if he does both of them, that is better and is more on the safe side, and after that he may bow; so he may say the first takbeer whilst standing then say takbeer whilst leaning forward for bowing. This is the most complete way. But if he does not say the second takbeer and limits it to the first one, that is sufficient according to the correct view. End quote. 

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him)