change asp cookie value using aspx(C#)

Hi I am reading asp cookie using c#.net:

string name = Request.Cookies["name"].Value; (OK)

I am trying to change the value in asp cookie, after the user clicks on the button (goes to another aspx file),

Response.Cookies["name"].Value="":    or Response.Cookies["name"].Value=null; 

IT's wired. When I get to the second aspx file (where I change the cookie value) and when I print the value to a file it prints the new value. But when I check the value using javascript in the browser javascript: alert(document.cookie); it displays the old one. and when I click back on the browser  and check the cookie value it prints the old value.. (is there away to set the asp cookie's value to null ? using c#.net)

 

Is there a different way to change a value in asp cookie using aspx (C#)
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