Formula for calculating the local point scale factor of a secant Mercator projection with Autochart

I am looking at a projected coordinate system with the following parameters:

• Datum WGS-84
• Spheroid WGS-84
• Semi-Major Axis (a) 6 378 137.000 m
• Inverse Flattening (1/f) 298.2572235634
• Latitude of Origin of Projection 0°N
• Longitude of Origin of Projection 25°W
• False Easting (metres) 0 000
• False Northing (metres) 1 000 000
• Standard Parallel 25°N
• Scale Factor 1.00000 at Standard Parallel

When using Autochart for Autocad (Inquiry –> Point Id) when I click at a point, I get a Scale factor value.

For 25° N and 25° S, the scale is 1.000, for 0° N it is about 0.90685. It increases above 1 towards the poles.

From googling I understand that this is a secant Mercator projection, where two parallels offset from the equator have a unity scale factor.

I am trying to find the proper formula to calculate this for a given lat/lon point, but the fact that I also get values below 1 mystifies me.

In this answer the graph clearly shows the scale factor increasing with the latitude and always being above 1.

This answer also provides formulas, but they still refer to the case where the equator is the latitude of origin.

What I don’t see in these formulas is where I would specify what my latitude of origin is, so I could then also solve for the case between the two secant parallels.