iMac login screen only half displayed
I'd start with the standard "troubleshooting toolkit" ...
do all, in this order.


Reset the SMC (instructions depend on Mac model)
Reset the NVRAM


At the boot chimes, hold  Cmd ⌘   Opt ⌥   P   R  
Keep holding all 4 keys until you hear the chimes again, then release

Verify &amp; if necessary Repair disk, using Disk Utility.
Fix permissions.
Before El Capitan this could also be done from Disk Utility, now it needs to be done from Terminal.
Verify permissions
sudo /usr/libexec/repair_packages --verify --standard-pkgs /
Repair permissions
sudo /usr/libexec/repair_packages --repair --standard-pkgs --volume /
Reset Home folder permissions &amp; ACLs
This has also become more complex since El Capitan because of System Integrity Protection, but is still possible by booting to Recovery Mode...  


At boot, hold  Cmd ⌘   R  at the chimes
At the recovery screen, open Terminal from the Utilities menu.  
Type in resetpassword and hit Return\Enter. A Reset Password window opens.
Select your username from the drop-down menu labeled Select the user account (NOT System Administrator/root).
Click the Reset button at the bottom of the window in the Reset home folder permissions and ACLs section.
Quit the Password Utility and go back to the main recovery screen.
On your keyboard, press  Cmd ⌘   Q   and restart your computer (or Select   > Restart from the menu bar). It's very important that you don't hold down the power button to exit the recovery session, or the ACL reset won't be performed.

Start in Safe Mode 


Start or restart your Mac. 
Immediately after you hear the startup sound, press and hold  Shift ⇧  
Release  Shift ⇧   when you see the Apple logo appear on the screen.
Leave a few minutes after login for caches to rebuild, then reboot normally.

Update to - or if already installed, then re-apply - the latest available combo updater for your current OS, not the usual delta update from App Store.  

Note: You can always search the latest combo update at http://www.apple.com/us/search/os-x-combo-update?src=serp &amp; hone further by adding the version you're looking for in the search box at the top.