What is the ruling on slaughtering a cow one day after giving birth, fearing its death, and is it true that it is in a state of postpartum impurity and it is not permissible to slaughter it until after forty days?

We do not know of any basis for what was mentioned about prohibiting the slaughter of an animal that has recently given birth, considering it to be in a state of parturition (nafsāʾ), nor does it have a special ruling. However, it is forbidden to slaughter milk-producing animals without need, as in the Prophet's (peace be upon him) saying to the Ansari: "Beware of the milch animal," due to the waste of milk involved therein. This prohibition is for guidance, not for outright prohibition (taḥrīm). The slaughter of young camels and cattle, and milk-producing sheep, may be prohibited for the public good.