using vectors is faster . . . .
Do you understand that "between" requires two things? The angle between the diagonal and what?Prove that the angle between the diagonal of a cube is not dependent upon the edge length of the cube.
One thing you might try: Set up a coordinate system with one corner of the cube at (0,0,0) and the others at (s, 0, 0), (0, s, 0), (0, 0, s), etc. That's similar to pacman's suggestion that you use vectors.
If this is a trig-class (as it's in the Trig Forum Section. . .), they really aren't looking for a "proof", so much as you reasoning out an example. An approach would be to be one side of the cube and place it on the normal x,y plane and go about solving for the diagonals. You would have to use some reasoning (drawing a diagram would help) to show this for diagonals moving in 3 spaces but it can be done (assuming you are not familiar with vectors, as has been assumed here).