When the song "All too well" was played on TV, my cousins asked themselves the following question: Is this a piece of theatre of a live braodcast?

Correct sentence - When the song "All too well" was played on TV, my cousins asked themselves the following question: Is this a piece of theatre of a live broadcast?

Direct speech does exactly what its name implies: it asks questions directly. Note that in this construction, the entire statement consists of the question itself.

When the song "All too well" was played on TV, my cousins couldn’t tell whether/if it was a piece of theatre or a live braodcast?

Correct sentence - When the song "All too well" was played on TV, my cousins couldn’t tell whether/if it was a piece of theatre or a live braodcast.

In indirect speech, however, a question is embedded in a longer sentence, and no question mark is used. Very often (but not always), this construction involves clauses beginning with whether or if.

David woke up from the coma and asked, Why am I still in the hospital.

Correct sentence - David woke up from the coma and asked, Why am I still in the hospital?

Direct speech does exactly what its name implies: it asks questions directly.