Of course they can see it again once they refresh the page but is there any way using which I can find using JQuery or Java Script that FormsAuthentication cookie has expired
Yes you can. Phil Haack wrote a very nice blog post explaining how you could prevent the Forms Authentication module from simply redirecting to the login page but instead return a 401 status code which your client AJAX call might catch and use window.location.href
to manually redirect to the login page.
He also wrote a NuGet called AspNetHaack
which you could use and if you are interested browse the source code here: https://github.com/Haacked/CodeHaacks/tree/master/src/AspNetHaack