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

                What is the ruling on one who finds the imam in the process of rising from bowing, so he says the takbeer to start praying (takbeer al-ihraam), and includes the takbeer for bowing in it. Is his prayer valid or not?.
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)