In reality, though they are actively fighting against the darkness, they are serving it by attacking guardians, the current champions of light. Not a fallen are bad though. For instance, in one of the Grimior cards Cayde is fighting a horde of hive with a fallen caption. They work together to kill the hive for quite some time. Unfortunately at the end the Captain attempts the kill Cayde (she had no strength left and thus it was a feeble attempt). However, it shows that fallen and guardians can work together.
English
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It is suspected that the fallen knew she was going to die due to her injuries. But suicide would be the coward's way out. The Fallen captain wanted a warrior's death, so she acknowledged Cayde-6 as an honorable warrior and gave him a reason to kill her.
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Very likely and most probably true.