Is it possible to trace the password the application is attempting to use to connect? I have an application attempting to connect to SQL Server. The application connects using a SQL Server account. The password for this account was accidentally changed. Now the application is attempting to connect to SQL Server using the wrong password. 

Login failed for user ''. Reason: Password did not match that for the login provided. [CLIENT: ]

Ideally, we can just change the account/password the application is using to connect. If this were not an option, is it possible to trace the password the application is attempting to use to connect?
The short answer is you can't see the password being used to connect.  That being said, you can restore a backup copy of the master database to a test instance, get the hashed password, and then apply it with the HASHED option of ALTER LOGIN.

The password hash can be retrieved from the restored master database with:

SELECT LOGINPROPERTY('your-sql-login', 'PasswordHash');


The password can then be applied to the other instance, substituting the value from the script above:

ALTER LOGIN your-sql-login
WITH PASSWORD = 0x01000CF35567C60BFB41EBDE4CF700A985A13D773D6B45B90900 HASHED;