This is the 5854th most frequent Japanese word.
ふるう
To exert, wield, or display (strength, ability, talent, etc.).
In this sentence, "ふるう" is used to mean "to display" or "to exert," referring to the showing or using of something like courage or power.
彼は危険な状況で勇気をふるうことができた。
He was able to display courage in a dangerous situation.
Here, "ふるう" again implies the exertion or demonstration of a quality or ability, in this case, calmness.
彼女は冷静さをふるうことが必要だった。
She needed to maintain her calmness.
In this context, "ふるう" contains the nuance of excelling or showing exceptional quality.