どうして・なぜ
why
どうして / なぜ + question
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どうして and なぜ both ask "why". どうして is the everyday word and sounds a little softer; なぜ is more formal and common in writing. Among friends you'll mostly hear なんで.
Answers usually give a reason with から: どうして休みましたか。——風邪を引いたからです, "Why were you off? Because I had a cold."
どうして on its own is a complete question: どうしてですか, "why is that?"
Be careful with なんで: 何で can also mean "by what means" (何で行きますか, "how are you getting there?"), so the polite question "why" is safer as どうして.
Register: どうして is everyday; なぜ is more formal or written; なんで is casual.
Examples
どうして昨日休みましたか。
Why were you off yesterday?
どうして日本語を勉強していますか。
Why are you learning Japanese?
どうして来なかったの？
Why didn't you come?
なぜ空は青いのでしょうか。
Why is the sky blue?
どうしてですか。
Why is that?