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Even though you may be allowed to claim one of the children as your dependent per the terms of your divorce agreement, for purposes of claiming the Child Care Expenses, only the custodial parent with whom the child lives for most of the year may actually claim the expenses for child care.
The purpose of the child care credit is to allow someone who has a child or other qualifying dependent to be able to pay for their care while they work or look for work, and so the credit is not allowed to be claimed by the non custodial parent, even if that parent is allowed to claim the child as a dependent due to a divorce agreement.
You can refer to pages 3 to 5 of the following IRS publication which covers this topic, under the paragraph entitled "Children of Divorced or Separated Parents".
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