Do I need to file Form 8332?

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If you are the custodial parent and wish to release the claim of exemption to the non-custodial parent, then yes. This allows the other parent to claim the dependency and Child Tax Credit, but not the HOH filing status, EIC or Child Care Credit.

What is Form 8332: Release/Revocation of Release of Claim to Exemption for Child by Custodial Parent 

Having custody of your child usually means you can claim an exemption for that child as a dependent on your taxes. But if you don't have to file a tax return, or you reach an agreement with your child's noncustodial parent, you can let them take the exemption instead with Form 8332.