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In Episode 3 Revenge of the Sith Palpatine/Sidious strings Anakin along to the dark side with the promise that together the two of them will be able to re-discover Darth Plagueis' secret to cheating death.
"To cheat death is a power only one has achieved, but if we work together, I know we can discover the secret."
From the end of Revenge of the Sith to their deaths in Return of the Jedi did Darth Sidious and Darth Vader ever manage to re-discover this secret?
Palpatine does, to an extent. He preserved his essence after DS2 was destroyed at Endor and took over bodies of specially designed clones.
This is covered in detail on Star Wars Wikia's Palpatine article, start reading from "Palpatine Reborn" section.
At an unknown point in time prior to his death at the Battle of Endor Palpatine had found a different way to cheat death by preserving his spirit after the death of his body, by the method of spirit transference. He arranged for a series of clones of himself to be created for his spirit to possess, in the event that he would perish. His primary supply of clones was kept on Byss, guarded by loyal Dark Jedi and immense genetically-altered guards and maintained by a trusted private physician. Unfortunately, the clones were heavily ravaged by the dark side and not sustained by the Force. Each body he took would age and deteriorate more quickly than the last. But the Dark Lord wasn't concerned; he had an endless supply of clones which he could use to rule the Empire.
After the destruction of the second Death Star, Palpatine's spirit was forced to journey in the maddening, bodiless existence of the void, eventually taking possession of the body of Emperor's Hand Jeng Droga. Palpatine called out to Sate Pestage, who rescued the broken body from Kaal and returned it to Byss. Though Droga went mad in the process, he was able to journey to Byss, where Palpatine was able to take possession of a clone of himself. There, on Byss, he would remain for years and would rebuild his strength and Empire. Resurrected, Palpatine planned to replace the Galactic Empire with a "Dark Empire", a universe-spanning magocracy ruled through the dark side of the Force alone, without the need for regional governors or technological domination.
Most of this is sourced from Dark Empire comics
As far as Vader, the answer is both yes and no. Vader the Sith Lord did not. Anakin Skywalker the Jedi did, in a way, since he managed to learn to communicate as a Force Spirit (as evidenced by him appearing with Obi-Wan and Yoda to Luke at the very end of ROTJ).
Going off the novel that bears his name, it is safe to say that Darth Plagueis' secret died with him and wasn't rediscovered by his pupil:
Plagueis had discovered a way to influence a lifeform's midi-chlorians to either kill an individual (as he did with King Veruna) or to prolong their life (as he did with himself). As far his ability to bring people back from the dead, he performed this feat several times in his experimentation with Darth Venamis.
The canon novel Tarkin reveals that Sidious intended to continue Plagueis' research (with Vader's assistance) once the Death Star was completed, and the Empire demanded less of his attention:
The sooner the [Death Star] is completed, the sooner you [Vader] and I [Sidious] can devote ourselves to more pressing matters - matters only you and I can investigate and that have little to do with the Empire."
[...]
Save for Sidious, no sentient being in close to five thousand years had set foot in the [Sith shrine beneath the Imperial Palace]. The room's excavation and restoration had been carried out by machines under the supervision of 11-4D. Even Vader was unaware of the shrine's existence. But it was here that they would one day work together the way Sidious and Plagueis had to coax from the dark side its final secrets.
[...]
As powerful as the battle station might become, its real purpose was to serve as a tangible symbol and constant reminder of the power of the dark side, and to free Sidious from having to portray that part.
[...]
Attacks like the one directed against Tarkin's moon base and discoveries like the one on Murkhana were distractions to his ultimate goal of making certain that the Force could not strike back
Tarkin Chapter 9: "As Above, So Below"
Tarkin is set five years after Revenge of the Sith, so evidently no research was being done during that time.
Since the first Death Star was not fully operational until the events of A New Hope, it also seems doubtful that there was any research happening in the fourteen years between Tarkin and that film; without a Death Star, and with an Imperial Senate, Empire-running must still occupy the bulk of Sidious and Vader's effort. According to the best estimates we have, the Death Star was destroyed mere days after the destruction of Alderaan.
Since Sidious is evidently delaying the commencement of their research until the completion of the first Death Star, it also seems likely that he'd delay it until the completion of the second Death Star, which he clearly intended to crush the Rebel Alliance and restore "harmony" to the Empire. Since this didn't happen, it seems most likely to me that he and Vader simply never got around to teasing out the mysteries of the Force.
From The Rise of Skywalker, the answer is clearly
Yes as Palpatine/Sidious has sort of resurrected himself at the beginning and uses Ben's and Rey's power to fully resurrect himself at the end of the movie.
If one accepts as canonical the Dark Empire comic books, then yes, Sidious does.
He transfers himself at death into prepared clones by a perversion of the same technique that Obi Wan had used to become one with the force.
This is the same volume where Luke then follows in Anakin's footsteps, becomes Sidious' apprentice, and has to be turned back by another jedi... Fortunately for him, he'd trained one.
The idea of "cheating death" is very ambiguous considering that a Jedi's Force ghost essentially "cheats death" by its mere ability to speak to the living. In Yoda's case, not only can he commune with the living, but he can also tap them on the head with a walking stick, and shoot lightning bolts out of the sky. The Sith lords' idea of "cheating death" has to do with maintaining their worldy forms, whereas the Jedi cheat death by becoming Force ghosts.
