Upvote:2
Yes, your cousin's valid US business visa allows this (I presume it is a B-1 visa, or a combination B-1/B-2).
This is clear from the State Department's page on transit (C) visas:
If you already have aΒ valid visitor (B) visa, you may be able to use it to transit the United States.
The reason they say "may" is because they have in mind the fact that no visa guarantees that the traveler will be admitted by the border officer.
When making the transit through the US, your cousin will need to pass through both immigration and customs controls. Before customs, he will need to retrieve his luggage, and after customs he will need to re-check it.