The rules for claiming a step child are the same as the rules for claiming a biological child even if you do not adopt the step child, that legal relationship is also not dissolved if you divorce the other parent.
However, the only parent who has the automatic right to claim a child as a tax dependent is the parent where the child sleeps more than half the nights of the year. The IRS only follows where the child physically lives, and does not follow court orders regarding custody. If the child does not live in your home, you cannot claim the child as a dependent unless the parent who has actual physical custody more than half the nights of the year gives you a signed release form.
