Can you shadow your relatives?

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It will be good for the experience and shadowing hours but I don't think you can get a letter from him/her
 
Its fine as long as it isn't your only healthcare experience. Like other posters have mentioned, this person cannot send an LOR but i think it helps.
 
You might want to ask your cousin if he has any other DO friends at work or in his network that would be willing to let you shadow them-- that way you can get the needed experience and possibly the letter. I say this as someone with no DO connections--if you have it, use it!