I've wondered that too, I hate that.

The first amendment lets them use the word god as long as they don't favor one god over the other. Can't say Jesus, Allah or Yaweh.

As I see it, governments are religions. Religions are basically entities willfully ignorant of reality. Our government prefers popular opinion over reality.

There is a prayer because, despite the First Amendment, the Christian religions are the defacto state religion. Note that "In God we trust.", uses a capital "G" not lower case, indicating "the one true God, A.K.A., Christian god. "Do you swear to tell the truth...so help you God?" also uses the upper case "G". As any English teacher will tell you, the upper case "G", as opposed to lower case, refers to "The" God. Zeus, Aten Geb, Isis, Khnum, and Seth, et. al. would not be referred to as upper case, "God". Therefore, defacto Christian state religion.

Perhaps because the framers and courts, in their hopelessly Christian indoctrinated cultural frame of reference feel at a gut level that "religious" freedom means you can pick any brand/denomination of Christianity you like.

because its not true

yeah - about that. We are the same, our parliamentary sessions begin with prayer.