On a more analytic note:

arctan x = x – (x^3)/3 + (x^5)/5 – (x^7)/7 + (x^9)/9 – ...

If we take any portion of x, the tangent of the angle, we cannot be taking the same portion of the angle (the value returned by the formula).

Or, looking at the Formula for the Tangent of an Angle

Again, dividing the angle, x, by any number does not produce the same fraction of the number produced by the formula (the tangent of the angle).

I think the geometric interpretations are the easiest to understand. Bisect or take any part of the angle and the geometry says it's impossible to take the same fraction of the tangent. Bisect the tangent and we cannot bisect the angle.

Hip-Valley Roof Framing and Joinery Math