In asp.net using Windows / forms authentication, you can find out if the user is logged in with:
<% Page.User.Identity.IsAuthenticated %>
Form authentication is cookie based, but can be configured to not use cookies with cookieless settings.
How to fully customize forms authentication is not a quick answer, but it is relatively easy to configure that there are many HOW-TOs online. Here is a good http://www.4guysfromrolla.com/webtech/110701-1.shtml
MSDN: http://msdn.microsoft.com/en-us/library/aa480476.aspx
You can also use Windows authentication, which does not use cookies.
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