にこたえて is a construction formed from に, and the て-form of the る-Verb 応（こた）える. It is often translated as ‘to meet (A)’, or ‘in response to (A)’. Like に応（おう）じて ‘to apply to’, にこたえて indicates that (B) is something that will change based on what is considered necessary, suitable, or appropriate on a case-by-case basis when it comes to (A).
にこたえて often appears in hiragana alone, and will attach to the end of nouns that it is marking as the (A) that something being done in response to.

• 生徒（せいと）達（たち）の要望（ようぼう）に応（こた）えて、レッスン料金（りょうきん）を1000円（えん）安（やす）くした。
  In response to our students’ request, we have reduced the lesson fee by 1,000 yen.
• クライアントのリクエストに応（こた）えて、デザインを少（すこ）しだけ変（かえ）えました。
  We changed the design slightly in response to the client's request.

Alternatively, にこたえる will be used when connected directly to a second noun.

• 弊社（へいしゃ）ではお客様（きゃくさま）のニーズに応（こた）えるサービスを提供（ていきょう）するよう、日々（ひび）努力（どりょく）しています。
  We are committed to providing services that meet the needs of our clients.
• あなたの要望（ようぼう）に応（こた）える自信（じしん）はありますが、このデザインを完成（かんせい）させるには時間（じかん）がかかると思（おも）います。
  We are confident we can meet your request, but it will take time to complete this design.

In many cases, (A) will indicate a request, wish, or expectation.