Is 你怎么知道怎么做中国菜? (How do you know how to cook Chinese food?) a grammatically correct question?

Question

This is from a problem from my Chinese book; I'm practicing the use of 怎么 (zěn me).

美国朋友会做中国菜，你觉得很奇怪，问他：...
_{Měiguó péngyǒu huì zuò zhōngguó cài, nǐ juédé hěn qíguài, wèn tā: ...}

This translates to "An American friend can cook Chinese food; you feel it's strange. [You] ask him: ... " I came up with two answers:

The first I think is fine: 你怎么会做中国菜？ (Nǐ zěnme huì zuò zhōngguó cài?) meaning "How can you cook Chinese food?"

My second answer I'm uncertain about because of the repeated 怎么, and because it's an almost-direct translation from English:

你怎么知道怎么做中国菜？
_{Nǐ zěnme zhīdào zěnme zuò zhōngguó cài?}

By this I intend to ask him "How do you know how to cook Chinese food?"

Question: Is 你怎么知道怎么做中国菜? a grammatically correct question?

Answer 1

It is grammatically correct, but it looks not nice. Or you can say that:

你怎么知道中国菜的做法？

Answer 2

你怎么知道如何做中国菜？

or without how

你在哪学会做中国菜？

Answer 3

Both of your sentences are grammatical. The second one is wordy though. If you want to express surprise 奇怪, you can put 你竟然会做中国菜？if you only want to practice 怎么，your first sentence is fine.

Answer 4

Your Chinese is a bit awkward in this case as you suspected. I would use

你怎么会做中国菜?