Single Sign On for Main and My Site

I don't know if this is the right place to ask but here is my question:

I have two sites, one  is my portal called http://site and the other is My Site for my users called http://mysite, internally they are working fine and the users don't need to perform a typical user log on with password and user name.

I don't know if it is possible but here is what is on my mind, on the extranet users are required to log on by providing their user name and password to gain access to the websites, and the big hassle is they need to log on twice, one time for each, and I would like to know if it is possible to have some mechanism to capture the first log on and share to the other website, if user logs on to http://mysite, the same credentials will be used (automatically) and the user gets logged to http://site when browsing to this location without the need to re-input again the credentials.

Externally both sites are secured by HTTPS, and I'm using SharePoint 2010 with my own Claims Authentication mechanism, I don't know much about custom providers and stuff like this but in SharePoint 2007 there was Single Sign On and I'm using this term "Single Sign On" as my "catch phrase" here to illustrate what I'm trying to accomplish :)

Thanks!!!