〜てもともと
nothing to lose (if it fails)
Verb て-form + もともと · Noun / な-adj + でもともと
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てもともと says that failing would just leave you where you started, so you might as well try: 断られてもともとだから、頼んでみる, "I've got nothing to lose if they say no, so I'll ask".
もともと (元々) means "originally, from the start". So the logic is "even if it fails, I'm just back at zero". It's encouraging and upbeat.
The most famous version is だめでもともと, often shortened to ダメ元: ダメ元で応募してみた, "I applied, figuring I had nothing to lose". The second half is usually an attempt: やってみよう, 頼んでみる, 受けてみた.
Don't confuse it with てもいい, "it's okay to".
Examples
失敗してもともとだ。挑戦してみよう。
We've got nothing to lose. Let's give it a try.
断られてもともとだから、頼んでみる。
I've got nothing to lose if they say no, so I'll ask.
負けてもともとの気持ちで戦った。
We played with nothing to lose.
落ちてもともとだと思って、試験を受けてみた。
I took the exam, figuring I had nothing to lose if I failed.
だめでもともと、やってみよう。
We've got nothing to lose. Let's go for it.