It is better (1)/to keep one's head in the face of danger than (2)/losing one's courage (3). / No error (4)

Answer

3

Explanation

to lose/lose will replace losing because - the basic pattern is as follows:

Look at the examples given below: It is better to be hated for what you are than to be hated/ be hated what you are not.
It is better to beg for forgiveness than to ask/ask for pernission
Hence, to lose/lose one.s courage is the right usage