I have Sharepoint 2013 farm which has a two way trust relationship with another domain.
The user is forced to type the domainname\user to login .
IS there a way that the user does not need to enter the domain name in Internet explorer and just enters the User name
This behavior is not unique to your domain trust configuration, it's this way with traditional domain deployments as well. Typically when you see the logon prompt it will show you the domain you are trying to logon to. If this is already the correct domain you don't have to prefix (or use UPN). If this is not the correct domain then you must specify it as part of the logon process.
If the users use a domain account to logon on to their windows machines you can add the URL to their intranet zone (or any zone if you configure it to allow automatic logon). This does not remove the domain requirement but will pass it silently in the background so the users doesn't need to see the logon prompt.
The other method is to enable forms based authentication. This will allow you to control the authentication inputs - however you can't use the automatic logon functionality as stated above.
For non-windows based machines like MAC you can configure password helper like KeyChain to store and forward the credentials. The password vault in windows would work in a similar fashion.
I am thinking two solution for this scenario.
Basic authentication enables a web browser to provide credentials when the browser makes a request during an HTTP transaction. Because user credentials are not encrypted for network transmission but are sent over the network in plaintext, we do not recommend that you use basic authentication over an unsecured HTTP connection. To use basic authentication, you should enable Secure Sockets Layer (SSL) encryption for the web site; otherwise, the credentials can be intercepted by a malicious user.
