For example, there is a sentence in the textbook Nuovo Progetto Italiano:
Uno di voi è l'agente di polizia che cerca di verificare quello che è scritto nell'agenda del ragazzo.
Can I use "che cosa" instead here? Thank you!
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They're not the same in this case and generally.
There is a person speaking to other people, among which there is a police officer in charge of verifying something that's in the boy's diary (appointment book).
What's written is already a piece of information that only needs to be checked. For instance there could be written “Meet X at 9:30” and the officer should check whether the meeting actually happened in order that it provides an alibi for the boy.
The difference is that quello is a demonstrative pronoun, whereas che cosa is an interrogative pronoun. They serve different grammatical and logical purposes.
Let's try a simpler example:
Prendi il taccuino e dimmi quello che c'è scritto.
Prendi il taccuino e dimmi che cosa c'è scritto.
Are they different? Yes. In the first case I know that something is written in the notebook, in the second case I don't know whether it has any notes. Actually, this is a case where the difference is not so important and both forms mean essentially the same. But it's not in the case of the police officer, because of the action involved, namely verificare.
In this case "quello che" can be substituted for by "che cosa", "cosa", "che", and also by "quel che", "ciò che", "quanto". It's only a matter of style. You can consider the first set of alternatives as more informal, ordered by informality. Some may even say that "che" is regional when used in this way. On the contrary the second set can be considered as more formal in order of ascending formality.
For clarity from the highest level of formality to the lowest: "quanto", "ciò che", "quel che", "quello che", "che cosa", "cosa", "che"
With any of these alternatives, it's not taken for granted that anything is written at all, even though the speaker assumes that something is written. One can express doubts about this circumstance by adding a remark like the following one, even without changing "quello che".
Uno di voi è l'agente di polizia che cerca di verificare quello che è scritto nell'agenda del ragazzo, sempre che ci sia scritto qualcosa.
If you say instead "quella cosa che" ("that thing which") you are referring to something you and your listeners already know about.
...verificare che cosa è scritto..., or
...verificare che cosa c'è scritto...
The second form is probably more common, the particle "ci" (the i is dropped by the apostrophe in c'è) means "there".
Note the masculine scritto: it could be feminine, because cosa is feminine, but not often used (this is not a simple argument).
The particle che could also be substituted by quale:
...verificare quale cosa è scritta..., or
...verificare quale cosa c'è scritta...
and in this case scritta has to be feminine because the value of cosa is clearly and exactly a noun (this value of noun is quite obfuscated in the two first phrases: che cosa can be rendered with "what" in English, while quale cosa can be rendered with "which thing")