Whose post are you replying To?
What is the focal length of the scope you are using? It looks to me like you're computing the field for the 8 inch F/6. This provides a 6.83 mm exit pupil, the OP was asking about 5mm and 6 mm exit pupils.
These are small angles, the small angle approximation is plenty accurate.
The half tangent provides 2.1615°
The small angle approximation provides 2.1619°, that's an error of 0.015%
For that to be significant, the field stop would need to be accurate to 8 microns on the field stop..
The false claim that the formula is exact is problematic.
Its bad practice to claim a formula to be exact when it isn't .
This only leads the inexperienced to apply it to situations where it is wildly inaccurate. (there are many examples on various websites where the author has assumed that a formula is exact when it isn't and even done a series expansion which is of little or no value because the formula from which the series was derived is inaccurate).
The assumptions made should always be clearly stated so that an approximate formula can be used appropriately.
e.g. for half field angles of less than 0.1 radians (5.7 degrees) the error is less than 0.33% and increases approximately in proportion to the square of the semi field angle.
The actual difference in the example is approximately 0.012% not 0.015%.
The error that results from assuming 1"= 25mm is more significant.
Care should be taken when the field stop is internal such as for eyepieces employing a Smyth lens in front of the field stop.
Similarly where the exit pupil has significant SA, formulae using the exit pupil diameter can be wildly inaccurate.