One of the common misconceptions that divorced parents have is that whatever the divorce agreement says will be honored by the IRS.  This is no longer true.  Starting in 1996, the IRS came up with the custodial parent rules and Form 8332 (or its equivalent) to determine who can legally claim a dependent.   

In brief, the custodial parent is the one where the dependents live more than six months out of the year.  The custodial parent has all rights to claim all credits (child tax credit, additional child tax credit, earned income credit, education credits, child care credits, etc).  In fact, the only credit that the custodial parent can give to the non-custodial parent is the child tax credit.  Let me use an example of how this currently plays out: