Regarding the Taraaweeh prayer (of Ramadhaan nights), do we say the Opening Du‘aa’ (supplication) after each Takbeerah (opening "Allaahu Akbar") or just one time in the first two Rak‘ahs? May Allaah reward you.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

The Opening Du‘aa’ is recited after every Takbeerah of Ihraam (beginning of prayer) with the exception of two places: (1) the Janaazah (funeral) prayer, in which there is no opening supplication, and (2) when someone joins the prayer in a position other than the Qiyaam (standing posture). But then once the Imaam ends those two Rak‘ahs and starts another two, the Opening Du‘aa’ is recited, and it is like that for every prayer.

The evidence for this is that the Prophet, sallallaahu ‘alayhi wa sallam, would open prayer (without specifying which prayer) saying: "Subhaanak Allaahumma wa-bihamdika wa-tabaarak ismuka wa-ta‘aala jadduka wa la ilaaha ghayruk. (Glorified are You, O Allaah, with Your praise, and blessed is Your Name, and exalted is Your Majesty. There is no god other than You)." [Abu Daawood, At-Tirmithi, An-Nasaa’i and Ibn Maajah]

An-Nawawi said in Al-Majmoo‘: "It is recommended for everyone who prays, whether an Imaam, a follower, someone by himself, a woman, a child, a traveler, one performing a Fardh (obligatory prayer), one performing a Naafilah (non-obligatory prayer), someone offering the prayer while sitting or while laying down, or otherwise, to recite the Opening Du‘aa’ directly after the Takbeerah Al-Ihraam."

It is known thereby that the Opening Du‘aa’ is not said in the second Rak‘ah, but only in the first.

Allaah Knows best.