The simple conic projection was once a very popular projection in atlases, particularly for maps of countries like Canada:
as this map shows, having a standard parallel of 55 degrees North latitude, appropriate at least for the most populated portions of Canada.
As the diagram below illustrates:
the cone contacting the standard meridian, when flattened to a sheet of paper, contains the standard meridian as an arc with radius cot(lat_sp), which is cos(lat_sp)/sin(lat_sp). Meanwhile, the standard meridian itself on the globe has radius cos(lat_sp), and its scale is true on the projection.
Hence, on a conic projection, the meridians project out from the center of curvature of the parallels at angles equal to the longitude multiplied by sin(lat_sp). Each parallel in the simple conic is a circle with radius cot(lat_sp)+(lat-lat_sp), where lat and lat_sp are expressed in radians.