-고자 means "intending to" or "in order to" - formal purpose.
Formal intention - 알리고자 합니다 (I intend to inform)
Purpose - 성공하고자 노력한다 (strive in order to succeed)
Official tone - Speeches, formal writing
See examples below for varied real-world usage.
More formal than -려고
Often with 하다
Intending to inform you, I stand here.
I decided to study abroad intending to learn new technologies.
We are always making an effort with the intention of providing better service.
I arranged this meeting in order to listen to your opinions.
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