If you are looking to complete the words of power quests it is often hard to find Arngeir because he has no place that he always stays in. There are multiple locations that you can find him. These are:
In the beds (located at the end of the bottom right corridor).
Praying (located in the upper walkways of High Hrothgar, still inside. They can also be near the entrance, in front and behind the large stone).
Practicing his voice (either near the edge of the cliff to the left of the exit to the courtyard of High Hrothgar, or at the top of the tower in the High Hrothgar Courtyard).