The pop-up reason that Duolingo gave me for not accepting it is:
'We only accept "wife" if "Frau" is preceded by a possessive determiner.'
So, when answering questions like this on Duolingo, if there is no reference to some kind of relationship of 'Frau' to another person, Duo only accepts 'woman'.
In real life? I imagine it's correct, but just like in English, it would sound a little weird when spoken by someone other than the woman's partner.
Das Rezept is also used for a doctor's prescription. The following definition is from Duden "schriftliche ärztliche Anweisung zur Abgabe, gegebenenfalls auch Herstellung bestimmter Arzneimittel in der Apotheke" which translates as "written medical instructions for delivery, possibly also the production of certain medicines in the pharmacy" Duolingo cannot argue with Duden!
I am writing 'The woman is reading the recipe' but it is not accepting my answer
That's unusual - that sentence should be accepted in a translation exercise.
Do you have a screenshot of that sentence being rejected? If so, please share it with us -- upload it to a website somewhere (e.g. imgur) and tell us the URL of the image.
But when I checked Google translation for die Frau, it can also mean the wife.
Where do you see that? When I translate die Frau, I only see "the woman".
When I translate just Frau, I get several options, including "wife". And Frau can indeed mean "wife" in some contexts. But in die Frau without possession (as in die Frau des Präsidenten "the president's wife"), it means "the woman".
Also, don't trust Google Translate.