Explanation
Often used in the pattern 'Xこそすれ、Y〜ない', meaning 'One might do X, but definitely not Y'.
Formation
Verb-ますstem + こそすれ、(絶対に)～ない
Detailed explanation
Verb-ますstem + こそすれ is typically used to emphasize that while someone might certainly do or be X, there's no way they would do or be Y. The sentence that follows is usually a strong negative, denying the possibility of Y.
Examples
Kare wa tomodachi o tasuke koso sure, jibun o higesuru koto wa arimasen.
He might help his friends, but he never belittles himself.
Kanojo wa ryōri koso sure, sōji wa hotondo shinai.
She might cook, but she rarely cleans.
Watashi no ani wa benkyō koso sure, asobu jikan ga hotondo nai.
He might study a lot, but he barely has time to have fun.
Haha wa hataraki koso sure, jibun no tame ni jikan o tsukau koto wa hotondo nai.
My mother might work a lot, but she hardly spends any time on herself.
Practice
Grammar content sourced from hanabira.org.