The answers in this post are way too complicated for my taste.
Here's what I did:
Assuming you're on the branch you want to change
Step 1 - Make the branch the same as master
git reset origin/master --hard
Step 2 - Pull the branch that you want to be identical to
git pull origin branch-i-want-to-be-identical-to
Step 3 (optional) - Force push it to remote to overwrite the remote history
git push --force