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лёгкое
How do you pronounce the "ое" sound {IPA: [əjə]} at the end of a neuter nominative adjective such as "лёгкое"? I notice three different pronunciations:
-кое {with a weak "о" sound}
-ке {with the "о" sound dropped}
-коjе {with the "j" sound included}
Perhaps, the same goes for "прошлое", because the IPA is the same "[əjə]".
Examples in the Wiktionary: лёгкое, прошлое
The examples differ a bit with прошлое being pronounced with a more open А in -ае and audible Й -айэ, but this is idiosyncrasy of the speaker, i second the first example in your list
-кое {with a weak "о" sound}
Technically O must be reduced to A but due to the stress being a bit distant it's further reduced to something amorphous between Э and Ы, while Й of -йэ is reduced to short И → -иэ to keep final Е from devolving into Э
I'm not sure there is a "correct" pronunciation for those.
On paper, it's still [əjə], suggesting that it would be more "standard" to articulate the [j], but that seems fairly outdated because, by and large, nobody does, and dropping the [j] doesn't strike one as markedly colloquial the way, for example, dropping the [dʲ] in будет does. It's more of this ivory-tower approach of Russian academic linguistics, which often results in learners hearing different pronunciations from the ones they've been taught to expect, while rank-and-file native speakers are at a bit of a loss to explain what happened there.
I'd say the first one you listed is the most common and "neutral" one, but generally it's free variation.