“Thee” and “thou” in “The Lord of the Rings”

[Inziladun]

As for another LOTR occurrence of the "superior" speech, there are the Witch-king's words to Éowyn:

'Come not between the Nazgûl and his prey! Or he will not slay thee in thy turn. He will bear thee away to the houses of lamentation, beyond all darkness, where thy thoughts shall be devoured, and thy shrivelled mind be left naked to the Lidless Eye.'

Curiously, the Witch-king didn't use that language when he was face to face with Gandalf.

'Old fool!' he said. 'Old fool! This is my hour. Do you not know Death when you see it? Die now and curse in vain!'

It's "do you', not "dost thou". Does that indicate the Witch-king either didn't see Gandalf as an inferior, or that he was afraid?