〜に応えて
in response to, living up to
Noun (期待 · 要望 · 声援) + に応えて / に応え
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に応えて means acting in response to what others hope for or ask: ファンの声援に応えて、選手は手を振った, "the player waved in response to the fans' cheers".
The noun is usually 期待 (expectations), 要望 (requests), 声援 (cheers), 要求 (demands), 信頼 (trust) or リクエスト. 期待に応える, "live up to expectations", is a very common set phrase.
It's written with 応える (to respond), not 答える (to answer a question), even though both are read こたえる.
Compare に応じて, "in accordance with", which adjusts to a variable, and に反して, which goes against expectations.
Examples
ファンの声援に応えて、選手は手を振った。
The player waved in response to the fans' cheers.
両親の期待に応えて、医者になった。
Living up to my parents' expectations, I became a doctor.
お客様の要望に応えて、営業時間を延長した。
In response to customer requests, we extended our opening hours.
期待に応えられるよう頑張ります。
I'll do my best to live up to your expectations.
読者のリクエストに応えて、続編が出版された。
A sequel was published in response to readers' requests.