This is the 7836th most frequent Russian word.
казачьей
"Of the Cossack."
The word 'казачьей' is used here as an adjective describing 'отваге', meaning the bravery associated with the Cossacks.
Казак рассказывал о своей казачьей отваге.
The Cossack talked about his Cossack bravery.
'Казачьей' acts as an adjective here to specify the type of dance, which is a traditional Cossack dance.
На празднике все восхищались ее казачьей пляской.
At the celebration, everyone admired her Cossack dance.
In this sentence, 'казачьей' describes 'культуры', indicating the culture associated with the Cossacks.