Arrogant for the most part when confronting a second monster (10)
I believe the answer is:
cockatrice
'monster' is the definition.
(I know that cockatrice is a type of mythical creature)
'arrogant for the most part when confronting a second' is the wordplay.
'arrogant' becomes 'cocky' (I've seen this in another clue).
'for the most part' means to remove the last letter (most but not all of the letters).
'when confronting' becomes 'a' (I can't explain this - if you can you should give a lot more credence to this answer).
'a second' becomes 'trice' (both can mean a very short amount of time).
'cocky' with its final letter removed is 'cock'.
'cock'+'a'+'trice'='COCKATRICE'
Can you help me to learn more?
(Other definitions for cockatrice that I've seen before include "Monster" , "Device" , "fabulous creature" .)