Promise solemnly to use bad language (5)
I believe the answer is:
swear
I believe this is a double definition.
'promise solemnly' is the first definition.
(synonyms)
'use bad language' is the second definition.
(I've seen this before)
'to' is the link.
(Other definitions for swear that I've seen before include "Utter obscenities or take an oath" , "Promise on oath" , "Affirm" , "Curse" , "Eff and blind" .)