Deductions & credits

If she is the parent where the child physically lived for more then 1/2 the year then she is the custodial parent for tax purposes so she is correct.  ONLY the custodial parent when the child lived can claim the EIC and dependent care credit.  The other (non-custodial parent) can ONLY claim the child's exemption (dependent) and child tax credit.

You would enter the child this way to claim the dependent and tax credit but not the EIC or dependent care credits.

You should have a signed 8332 form (custody agreement) from the custodial parent that you must attach to a mailed tax return or mail it separately with Form 8453, (U.S. Individual Income Tax Transmittal for an IRS e-file Return) if e-filing. See Form 8453 and its instructions for more details.

This must be done within 3 days of your e-filed return being accepted by the IRS.
http://www.irs.gov/pub/irs-pdf/f8453.pdf

If you are the non-custodial parent where the child did not physically live for more than half the year (183 nights) then:

When you enter the dependent, you say that he is "Your child" (not you and new spouse if remarried),
He/she lived with you 0-5 months (less then 6 months),
“no” the child did not pay more than half of his/her own support,
"yes", you have a custody agreement, and
"yes” I am claiming this year.

That will give you the child's exemption, and child tax credit.  
 
The EIC, and Child Care Credits can only be claimed by the parent where the child actually lived.    


**Disclaimer: This post is for discussion purposes only and is NOT tax advice. The author takes no responsibility for the accuracy of any information in this post.**