Changing between Equirectangular and Stereographic projections?

[AlexSchacher]

I am trying to map out the basic landmasses of my geofiction world, from which I can zoom in on parts and go into detail with nicer looking maps. But I've hit a wall the past few months that has demotivated me quite a bit, and every time I come back to it I hit the same wall. So hopefully I can get this settled once and for all.

What I am trying to do is get my "master base map" in Equirectangular projection, from which I can use G Projector to convert to any other projection and base zoomed in maps from there. It works great for my purposes, but as Im sure you all know, G projector only imports Equirectangular, so converting back and forth doesn't work within the program. I am specifically trying to convert to and from Sterographic, so I am able to wrap my head around what I am drawing while mapping the polar regions, convert it back to Equirectangular and stitch the polar region back on. I have tried to use the program ReprojectImage but it doesn't seem to work right, or I am doing something wrong. Here are my images:

My world that I am starting with, in Equirectangular:
newworld2014_small.png

After converting this to Sterographic on the north pole:
test1.png

And after using ReprojectImage to try to bring this back to Equirectangular, and pasting it back on the master image. You can see my problem clearly:
failedattempt.png

So what it comes down to is: How can I switch back and forth between Equirectangular and Sterographic projections smoothly and accurately? And is there an easier way to map out my "master base map"?

EDIT: Found a solution myself, now I feel stupid for posting this here. I figured out how to use ReprojectImage correctly to bring a Sterographic back to Equirectangular.