Hun tager trøjen på i prøverummet for at se, om den passer.

On its own, tage usually does mean to take. But in this sentence it is part of the expression tage ... på, which with clothes means to put on.

So:

• tage = take
• tage en trøje på = put a sweater on

In this context, because she is in a fitting room and checking the fit, English would often naturally translate it as try on.

Because tage ... på is a separable verb expression.

In Danish main clauses, the verb part is often split like this:

• Hun tager trøjen på.

This is very common with verbs of clothing and motion.

A useful comparison:

• at tage trøjen på = to put the sweater on
• Hun tager trøjen på = she puts the sweater on

If the object is a pronoun, the structure is still split:

• Hun tager den på = she puts it on

So på belongs with tager, even though trøjen comes in between.

Trøjen is the definite form, meaning the sweater.

Danish often uses the definite form when talking about a specific item already known from the situation. In a fitting room, it is natural to say the sweater, not just a sweater, because we are referring to a particular one.

• en trøje = a sweater
• trøjen = the sweater

So Hun tager trøjen på means she is putting on a specific sweater.

They are close, but not exactly identical.

• tage trøjen på = put the sweater on
• prøve trøjen or prøve trøjen på = try the sweater on

In this sentence, Danish says she puts the sweater on in order to see whether it fits. That naturally implies trying it on.

So the sentence is focusing on the action of putting it on, while the next part — for at se, om den passer — explains the purpose: checking the fit.

Because Danish forms compound nouns very freely, much more than English does.

So:

• prøve = try / test
• rum = room
• prøverum = fitting room / changing room

Then the definite ending is added:

• prøverum = a fitting room
• prøverummet = the fitting room

English usually writes these as separate words, but Danish often combines them into one word.

i means in, and it is the natural preposition for being inside a room.

So:

• i prøverummet = in the fitting room

This is just like:

• i huset = in the house
• i bilen = in the car
• i rummet = in the room

for at + infinitive is a very common Danish structure meaning in order to ... or simply to ... when expressing purpose.

So:

• for at se = to see / in order to see

In the sentence:

• Hun tager trøjen på i prøverummet for at se ...
• She puts on the sweater in the fitting room to see ...

It explains why she is doing the action.

Here om means whether.

It introduces an indirect yes/no question:

• om den passer = whether it fits

So the idea is:

• for at se, om den passer
• to see whether it fits

Be careful: om can have other meanings in other contexts, such as about or in (about time), but here it specifically means whether.

Because om den passer is a subordinate clause, and Danish normally uses a comma before subordinate clauses.

So the sentence is divided like this:

• main clause: Hun tager trøjen på i prøverummet
• purpose phrase: for at se
• subordinate clause: om den passer

That is why you get:

• for at se, om den passer

This comma is very standard in written Danish.

Because trøje is a common-gender noun, and common-gender nouns are referred to with den.

• en trøje = a sweater
• therefore: den = it

If the noun were neuter, you would use det instead.

So:

• om den passer = whether it fits

Here den refers back to trøjen.

Because after a subordinating word like om, Danish uses subordinate-clause word order.

In a direct question, you would say:

• Passer den? = Does it fit?

But in an indirect question introduced by om, the subject comes before the verb:

• om den passer = whether it fits

So:

• direct question: Passer den?
• indirect question: Hun vil se, om den passer.

That difference in word order is very important in Danish.

Here passer means fits.

The verb passe can mean different things depending on context, such as:

• fit
• suit
• match

With clothing, it usually means to fit properly in size or shape.

So:

• Trøjen passer. = The sweater fits.
• Trøjen passer ikke. = The sweater does not fit.

In this sentence, she is checking whether the sweater fits her.