Hello,
Thank you for posting your question on Microsoft community.
Start and Cortana may not work due to corrupt system files.
I would suggest you to run system file checker(SFC) and see if it helps.
Refer this article:
Does System File Checker work in Windows 10?
http://windows.microsoft.com/en-us/windows-10/does-system-file-checker-work-in-windows-10
System file checker is a utility in Windows that allows users to scan for corruptions in Windows system files and restore corrupted files.
For Cortana, you may also follow these steps and check the issue.
a. Press Windows + R keys, type regedit in run
window and press Enter.
b. Navigate to the following registry location:
HKEY_CURRENT_USER/Software/Microsoft/Windows/CurrentVersion/Explorer/Advanced
c. Create a new "DWORD(32-bit)" value, type "EnableXamlStartMenu" and click
ok.
d. Open task Manager and close Microsoft Edge from there(if it is open ).
e. Restart the computer and check Cortana.
Note: Serious problems might occur if you modify the registry incorrectly. Therefore, make sure that you follow these steps carefully. For added protection, back up the registry before you modify it. Then, you can restore the registry if a problem
occurs. For more information about how to back up and restore the registry, click the following article number to view the article in the Microsoft Knowledge Base:
https://support.microsoft.com/en-us/kb/322756
I hope this information helps.
Please do let us know if you need any further assistance.
Thank you