If someone is your first cousin that means that you have a parent who is the sibling (brother or sister) of one of the parents of that person. Normally, this kind of relationship is easy to prove; your parents know your cousins parents and are directly related to them, so they will testify to that relationship if needed. But then, sometimes people are separated from their parents, perhaps their parents have died, perhaps they were given up for adoption at an early age. In such cases, genetic testing can show that a relationship exists, although it can't really narrow it down to being a first cousin. If you check people's birth certificates you can generally find out who their parents were. If they don't have birth certificates, the research is harder.
To prove a relationship to a first cousin you mustknow the parent through which that person is related to you, and how that parent is related to one of your parents. To prove a relationship to a second cousin or a more distant cousin, you must know the parent, grandparent and perhaps more ancestors.
Your cousin's cousin is no relation to you.
Your mother's first cousin is your first cousin, once removed.
Your mother's first cousin is your first cousin once removed.
Your mother's first cousin is your first cousin, once removed
The child of your first cousin is your first cousin, once removed.
The wife of your first cousin is not related to you.
Your Aunt's first cousin may be your second cousin, onve removed or may be no relation to you at all.
Your Aunt's first cousin may be your second cousin, onve removed or may be no relation to you at all.
Cousin once removed.
Either your second cousin or your cousin in-law
Your daughter and your first cousin are first cousins, once removed.
Your first cousin is your granddaughter's first cousin twice removed.