are both the inserted and deleted tables used in update trigger
Yes.
ex (table is empty to begin with)
INSERT INTO [dbo].[test_employee_table](letter)
values('a'),('b')
-inserted table count: 2
-deleted table count: 0
update [test_employee_table]
set letter = 'b' where letter = 'a'
-inserted table count: 1
-deleted table count: 1
delete from [test_employee_table]
where letter = 'b'
-inserted table count: 0
-deleted table count: 2