بسم الله الرحمن الرحيم
The abandonment of takbirul ihram when entering the prayer while the imam is bowing is a mistake due to the first takbir (which enters one into the prayer) being a pillar (of the prayer) which obligates the performance of it while standing. Afterwards the person enters the bowing position along with the imam (by saying a second takbir moving into the other position). The first takbir is separate from the (second) takbir for (entering) the bowing position. So the takbir for entering the prayer and then one for bowing is more complete and safe as Abu Huraira said, “whenever the Messenger -sallahu alayhi wa salaam- would stand for prayer he would say the takbir then he would say it (again) when bowing.”
Translated: Najeeb Al Anjelesi
Source: المنظار في بيان كثير من الأخطاء الشائعة