GIS, Mercator projection, calculate Y position

I am doing some stuff with GIS(or map), and I know how to calculate Y position from latitude in mercator projection. But I don't know how it comes?

Why it is ln(tan(x)+sec(x)) ? I couldn't 开发者_开发百科figure out why there is a 'ln'?

I know it is not directly related to programming, but I hope some one could help me understanding about this (better with some pictures to illustrate)

It sounds as though you're asking about the last step in the derivation you've listed. It's the definition of arctanh combined with some trigonometry identities.