There has been a lot of debate on Facebook about the question if a DX lens with a given focal length will produce a different image on a Nikon DX camera, than an FX lens with the same focal length on that same Nikon DX camera.
From a theoretical perspective, the simple answer would be no, the image will be about the same. There is even a mathematical equation to support this:

For a lens projecting a rectilinear image (focused at infinity) the angle of view (α) can be calculated from the chosen dimension (d), and effective focal length (f) as follows: α = 2 arctan d 2 f

α=2arctan(d/2f)

d represents the size of the film (or sensor) in the direction measured  (horizontal, vertical or diagonal).

That means that it doesn’t matter whether the lens is a DX lens or an FX lens. The is no room in the equation for DX or FX. Only minor changes will occur if both lenses will have other distortion. One lens could have barrel distortion and the other could have pincushion distortion.
On close focusing distances however, a lot of lenses (especially zoom lenses) “cheat” to be able to focus closer than one could expect from a lens with such focal length. The Nikon 70-200mm VR II AF-S G ED NIKKOR has an effective focal length of 135mm when zoomed in at 200mm and a focusing distance of 1.5 meter. This could explain why some people persist in their claim that a DX lens will give you an “extra reach” (or an extra crop factor) on a DX camera compared to an FX lens with the same focal length on the same DX camera. They simply focus to close to test this.
For “normal” focusing distances, there will be no considerable difference in the image.

 To test this, I used a Nikon D300s (DX) camera with the following set of lenses:

  • Nikon 18-55mm DX at 18mm, versus a Nikon 18-35mm FX at 18mm from 3.50 meter.
  • Nikon 35mm DX, versus Nikon 35mm FX from 5 meter.
  • Sigma 18-55 DX at 55mm and Sigma 55-200 DX at 55mm, versus Nikon 55mm FX from 6,7 meter.
  • Sigma 55-200mm DX at 200mm, versus Nikon 80-200mm from 14 meter.

I choose those zoom lenses because all the focal lengths were at the wide end or at the tele end of the lens (not somewhere in the middle). Not that that would make a difference, but I wanted to be sure that the focal length were at the fysical bounderies of the lens.
I used a tripod to ensure that the distance from the camera to the wall did not change when changing the lens.

My conclusion from the test: for “normal” focusing distances it doesn’t matter if a DX lens or an FX lens with the same focal length is mounted on a Nikon DX camera. The images produced will be roughly the same.

Count the bricks, and see for yourself.

18-55mm DX @ 18mm

18-35mm FX @ 18mm

35mm DX

35mm FX

18-55mm DX @ 55mm

55-200mm DX @ 55mm

55mm FX

55-200mm DX @ 200mm

80-200mm FX @ 200mm