Rikki shows he is brave when he goads Nagaina into fighting him after he kills Nag.
Rikki-tikki-tavi is brave because mongooses are too curious to be scared. Speaking about Rikki, the narrator says:
It is the hardest thing in the world to frighten a mongoose, because he is eaten up from nose to tail with curiosity. The motto of all the mongoose family is "Run and find out''; and Rikki-tikki was a true mongoose.
Rikki proves his bravery when he fights the snakes. A snake and a mongoose are natural enemies, and they both know it. Rikki was not frightened for long when Nag and Nagaina tried to kill him.
He was afraid for the minute; but it is impossible for a mongoose to stay frightened for any length of time, … and he knew that all a grown mongoose's business in life was to fight and eat snakes.
Thus in this altercation it seems as if the cobras are more frightened than Rikki. Rikki shows he is not frightened by chiding Nag for eating baby birds. After killing Nag, Rikki also chides Nagaina for not fighting.
"Yes, you will go away, and you will never come back; for you will go to the rubbish-heap with Nag. Fight, widow! The big man has gone for his gun! Fight!''
His intention is to get her away from the people, but he is clearly not afraid. Rikki is willing to fight Nagaina even though he knows she is dangerous. He got her off balance by killing her baby cobras in their eggs, but this also makes her mad with rage. To Rikki it matters not. He also follows her down into her hole, even though he knows that it is dangerous for him there. The point is to kill her, and he thinks nothing of the danger to himself.