No.
If you and the other parent live together, either one of you (but not both) may claim the child. You may decide between you which one will claim the child. Only if you can’t agree, do the IRS tie breaker rules apply, to see who has first choice. It may be worthwhile to prepare trial returns, both ways, to see which way the family comes out best. This tool may be useful: https://turbotax.intuit.com/tax-tools/calculators/taxcaster/?s=1.
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Even if you don't live together, you cannot do what you propose (one parent claims a child as a dependent and the other claim the child tax credit).
But, there is a
special rule in the case of divorced & separated (including never married) parents.
When the non-custodial parent is claiming the child as a
dependent/exemption/child tax credit; the custodial parent is still allowed to
claim the same child for Earned Income Credit, Head of Household filing status,
and day care credit. This "splitting of the child" is not available
to parents who lived together at any time during the last 6 months of the year;
then only one of you can claim the child for any tax reasons. The tax benefits
may not be split in any other manner.
Note in particular
that the non-custodial parent can never claim the Earned Income Credit, Head of
Household filing status or the day care credit, based on that child, even when the custodial parent has released
the exemption to him.
So, it's good idea to let the other parent know that you
will be claiming those items, as many first time divorced parents are not aware
of this rule and may try to claim those items, which will cause the IRS to send
out letters.
Ref: http://www.irs.gov/publications/p17/ch03.html#en_US_2014_publink1000170897
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