This code snippet cleared the cookies for my website upon loading my logout page.

if (!IsPostBack)
{
    // Clear cookies for Drawing Viewer upon logout
    if (Request.Cookies != null)
    {
        var cookies = Request.Cookies.AllKeys;
        foreach (var cookie in cookies)
        {
            var httpCookie = new HttpCookie(cookie)
            {
                Expires = DateTime.Now.AddDays(-1),
                Value = string.Empty
            };
            Response.Cookies.Add(httpCookie);
        }
    }
}