There is a belief that Vishwamitra and Kaushika are same person but then why would there be two pravaras of Kaushika and Vishwamitra in same gotra?
The pravara of Kaushika gotra is " Vaiswaamitra, Aghamarshana, Koushika " and the pravara of Vishwamitra gotra is " Vaiswaamitra, Devaraata, Owtala"
Technically, Vishwamitra is also called "kaushika" because he was the descendant of Rishi Kusha.
There was a king named Kusha, a brainchild of Prjaapati, and Kusha's
son was the powerful and verily righteous Kushanaabha. One who is
highly renowned by the name Gaadhi was the son of Kushanaabha, and
Gaadhi's son is this great-saint of great resplendence, Vishvamitra.
Vishvamitra ruled the earth, and this great-resplendent king ruled the
kingdom for many thousands of years. At one time the great-resplendent
king Vishvamitra went round the earth marshalling a unit of akshauhini
So King Gaadhi was the grandson of King Kusha and the father of Vishwamitra. Hence Vishwamitra is also known by the name "Kaushika" because his ancestor was Raja-rishi Kusha.
He was also a king but also a rishi. He is the rshi for the mantra of Rigveda. For example he is the rishi of the mantra 3.31
Hence Kusha was the propagator of Kaushika gotra and then again Viswhamitra was the propagator of Vishwamitra gotra.