For some odd reason Microsoft decided to remove the option to display the “Sign in as Different User” in SharePoint 2013.
The good thing is with just a quick change in the URL the you can get the site to reload with a log in box that shows for you to log in.
Hooray, no more Signing Out and closing the browser, and deleting your cache, and formatting windows and buying a new computer.
add the following string to the end of your URL:
Eg. http://SharePoint/ —> http://SharePoint/_Layouts/CloseConnection.aspx?loginasanotheruser=True
NB: This should only work on SharePoint 2013.