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Before Trunks became super saiyan rage (super saiyan hikari), he had a battle where he was able to hold, hit and take apart Black Goku super saiyan rosé from super saiyan blue Goku. How was this possible? We know by that moment Black Goku super saiyan rose was stronger than Goku Super saiyan blue, which is a transformation stronger than super saiyan god, which is a transformation stronger than SSJ3, which is a transformation stronger than SSJ2.

In the manga Toyotaro made SSJ2 Trunks to be "mastered" and become as strong as SSJ3 Goku, but in the anime, Goku SSJ3 Goku knocked with one punch SSJ2 Trunks (even though Beerus commented Trunks was a good fighter) . Did Trunks become much stronger after this, because of a zenkai boost or something? Did his sparring later with super saiyan blue Vegeta and the mental training he had made his SSJ2 much stronger (to the point where he could fight in SSJ2 against Black Goku SSJ Rosé) or is there any other explanation to the situation of SSJ2 being able to fight SSJ Rosé?

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Trunks in his SSJ2 state was definitely stronger than Goku and Vegeta SSJ2 in the Buu saga. However, weaker than SSJ2 Goku and Vegeta in super. I would say, his SSJ2 transformation is more or less relative to that of Caulifla's towards the end when she began to master it during her fight with Goku.
Trunks attained another transformation called Super Saiyan Rage during the course of battle with Goku Black. This transformation made him strong enough to at least hold his own against SSJR Black and made his power slightly relative to Blue. He easily surpassed SSJ3 Goku at this point. His power would be relative to current Mystic Gohan and Android 17 who are stronger than SSJG and more or less relative to SSJB tier(Not stronger).

As for trunks getting stronger, No, his fight with Goku or Vegeta didn't give him any Zenkai boosts. A saiyan attains a Zenkai boost when he is in a situation where his life is at stake. Goku merely took him out with one shot and Vegeta obviously didn't try to kill his son! The fight against Goku was Super trying to show where Trunks's power stacks up to the Z fighters now, as he hasn't come on for a very long time. His fight against Vegeta, was very likely a throwback to their training in Z and to create a father-son moment between a newly changed Vegeta(In terms of character) and Future Trunks.

We know the SSJ transformation is generally caused by rage. We've also seen powerups in SSJ transformations by characters when they are angry or someone's life is at stake. Example: Vegeta in his SSJ2 form against Beerus, Goku in his SSJB form against Black and Zamasu, Vegeta in his SSJB form against G.O.D Toppo. Trunks similarly, attained a very powerful rage transformation in this manner.

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Well, he was in a rage. It has been seen in the anime and movie (Battle of Gods), that when a Saiyan is enraged, their power increases manifold, usually you can understand it like the Hulk. We see Vegeta, in SSJ2 form, beat up Beerus for some time, and although Beerus wasn't serious, that's still an amazing thing to pull off.

So it is plausible to assume that in the moment of his enraged anger, he for some time, became stronger than Super Saiyan Blue.

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