You should clarify what you need, as what you request is how Sharepoint already works.
In order for the approver to approve an item he has to log-in to Sharepoint, using the built-in mechanisms. If you are using windows authentication (the default) instead of FBA, the user is logged in automatically using his Windows credentials. So there is no need (or point) for the user to re-authenticate - Sharepoint already knows who he is.
The only case where there might be an issue is when both requestor and approver have to use the same computer. Unless the requestor also has approval right, he will not be able to approve the request.
Are you perhaps concerned that a user with approval right may try to approve his own request? In this case you can prevent him from doing so by granting him read-only right to his request, ie removing the update right.