Simple answer: No.
But taxes aren't simple. So, maybe (answer at the bottom). Be forewarned, this is going to be long and complicated.
First, For tax purposes, there is no such thing as joint (50/50) custody, regardless of what your legal or informal agreement says. The requirement, to be custodial parent, is that the child live with you MORE than 50% of the time. One of you has to be the custodial parent and the other the non-custodial parent. The IRS goes by physical custody, not legal custody.
Yes, the IRS expects you to count the numbers of nights the child sleeps at each parent's home.
It is allowed for you to arrange the child's schedules so that the child spends more than half the year with the father one year and more than half with the mother the next year so that you are each the custodial parent in the year you claim the child, so that you can claim full benefits.
The custodial parent has first priority on claiming the children on her taxes; regardless of the amount of support provided by the non-custodial parent. Again, the IRS goes by physical custody, not legal custody. The non-custodial parent can only claim the child as a dependent if the custodial parent gives permission (on form 8332) or if it's spelled out in a pre 2009 divorce decree. Even if a divorce decree, dated after 2008, gives the non-custodial parent the right to claim the child, he must still get form 8332 from the custodial parent. A properly worded decree should require her to provide that form. https://www.irs.gov/pub/irs-pdf/f8332.pdf
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 dependency 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: https://www.irs.gov/publications/p17#en_US_2017_publink1000170897
Scroll down to "Children of divorced or separated parents (or parents who live apart)"
So, if you are the custodial parent, and the non-custodial parent is claiming the dependent this year, then yes, you can claim the dependent care credit.
