偵察するって英語でなんて言うの？

If you want to find out your girlfriends favourite food without her knowing then you could say 'you are being sly' or 'feeling out' these both mean you are doing it without the person knowing or as a secret so you could also say 'secretly'

The idiom sus out is from the word suspect and the phrase is used similarly to seek out.
On the down low is used by younger people and they are inferring the information is only spoken of quietly, or that the information is get down low, out of the eyeline of others.

On the sly means secretly/discreetly. If you want to do something on the sly, you do it without people noticing. You can use it in this situation and it can be positive. However, on the sly, can also be negative in a way of deceiving people. “He was having an affair on the sly” “He was stealing people’s ideas on the sly”.

To ask something "discretely" means to do so without trying to draw attention to the request.

Use of the word "note" in the second sentence implies trying to remember something without revealing you are doing it. A police officer would note a lie that a suspect tells them. A husband would note what items his wife is talking about when he is trying to select a birthday gift for her.

To ask something causally is similar to the use of the word discrete. To ask something casually, in this context, is to ask without drawing attention to the fact you want to know.

Surreptitiously = in a way that attempts to avoid notice or attention; secretively.
"Mary surreptitiously slipped from the room."
To go undercover = working secretly using a false appearance in order to get information for the police or government:
"An undercover police operation."
To spy on someone = to watch someone secretly so that you know what they do.
"Managers feel it is their duty to spy on the workforce."

>I want to invisibly find out what her favorite food is.
*Invisibly=in a way that cannot be seen or perceived.

>I am trying to find out her favorite food on the quiet.
*on the quiet=secretly.