Windows 10 starts out with the Cortana search bar taking up a fair amount of real estate on the Taskbar. I am not a fan of it just for all of that wasted space. You can simply right-click on the Taskbar and choose from the Search submenu on how you would like it to appear. You have the options of Hidden, Show Search icon, or Show Search box. This works nicely but there are a few instances where you might want to control this setting but you do not have the luxury of changing the setting for each user.
The typical answer to something like that would be to refer to Group Policy. Unfortunately, there is no Group Policy setting that controls how Cortana appears on the Taskbar. However, you can modify the Registry to set Cortana in your desired mode. You could deploy this Registry change with a startup script, a Group Policy Preference, or a roaming profile.
Open up Regedit and drill down to:
HKCUSoftwareMicrosoftWindowsCurrentVersionSearch
A DWORD value with the name SearchboxTaskbarMode controls how Cortana appears.
Set SearchboxTaskbarMode to 1 in order to get the search icon.
Set SearchboxTaskbarMode to 0 to remove the icon completely.
A value of 2 for SearchboxTaskbarMode sets the search to its default setting of the expanded search box.
Restart Windows Explorer after the Registry change and it will be reflected.