mysql error 1045 sqlstate 28000 access denied for user code example

Example 1: windows error 1045 (28000): access denied for user 'root'@'localhost' (using password: yes)

update user set authentication_string=password('my_password') where user='root';

Example 2: mysql error 1045

mysql> SELECT Host, User, authentication_string, PASSWORD('forgotten') FROM mysql.user WHERE User='nonexistant';  
+-------------+-------------+-------------------------------------------+-------------------------------------------+
| Host        | User        | authentication_string                     | PASSWORD('forgotten')                     |
+-------------+-------------+-------------------------------------------+-------------------------------------------+
| 192.168.0.1 | nonexistant | *AF9E01EA8519CE58E3739F4034EFD3D6B4CA6324 | *70F9DD10B4688C7F12E8ED6C26C6ABBD9D9C7A41 |
| %           | nonexistant | *AF9E01EA8519CE58E3739F4034EFD3D6B4CA6324 | *70F9DD10B4688C7F12E8ED6C26C6ABBD9D9C7A41 |
+-------------+-------------+-------------------------------------------+-------------------------------------------+
2 rows in set, 1 warning (0.00 sec)