[noun] + に応えて [noun] + に応え、[sentence] [noun] + に応えた + [noun]
に応えて means "in response to" or "living up to": someone acts to meet a request, a demand, an expectation or a need that others have directed at them. It goes straight after a noun, and the noun is nearly always one of a small set: きたい期待, ようぼう要望, ようせい要請, にいずニーズ, こえ声.
It belongs to news, business and formal speech, and the nouns it takes are formal ones. For an everyday request, conversation uses たのまれて頼まれて: ともだち友だちにたのまれて頼まれて, "because a friend asked me".
Both read こたえる. こたえる答える is answering a question; 応える is meeting what someone hopes for. Writing the wrong kanji is the common slip.
Before a noun, the pattern becomes にこたえた応えた or にこたえる応える. The stem にこたえ応え joins two clauses in writing.