How to get both sides of 1/2 inch water pipe into a new coupling
If you're dealing with copper, you will find two types of couplings.  One has a stop in the center and one without. What you need in this case is the type without a stop. Without  a stop, you can slip the coupling over the entire end of one pipe, and then slip it over the end of the other pipe.

With Stop (notice the dimple in the center)


(source: homedepot.ca)  

Without Stop

Without Stop http://www.usqualitysupplyinc.com/wp-content/uploads/2012/09/coupling-without-stop.jpg