Can i claim the daycare expenses i pay for a child of mine that does not live with me, court ordered i pay half of daycare expenses but i do not claim him on my taxes?

No.   Only the custodial parent can enter the childcare expenses and get the childcare credit.

Are you the custodial parent?  Do you have an agreement with the other parent to allow the other parent to claim them--due to divorce or that you live apart and share custody?  Did one of you sign a Form 8332?

If there is a signed 8332 then the custodial parent retains the right to file as Head of Household, get earned income credit and the childcare credit.  The non-custodial parent gets the child tax credit for children under the age of 17.

As far as the IRS is concerned, the custodial parent is the one with whom the child spent the most nights during the tax year--at least 183 nights.

If you are a non-married couple who live together then only one of you can claim the child(ren) and the one not claiming the child does not enter anything at all on their tax return about the child.