Ive already done a bit of research but this achievment is really glitchy and more than half of the people i asked could not get this achievment.
Others are 2nd mission, moa's, spoon full of blamite, target locater on the 3rd mission, high jack a banshee, firefight killionaire i can list prob 10 other achievments that people complained about...
+ its also gay when you play 45 mins into a mission then save and exit then come back a bit latter and beat it not getting the achievment.. bungie = fail

But if anyone knows what the exact problem is with the
If they came to hear me beg (25G) achievment - im all ears cause ive tried changing my armor,stopping the sprint in mid jump, different assassinations, different skulls, different difficulties, different #s of people in the game, different spots - pillar of autumn - tip of the spear , Restarted Xbox multiple times, Tried it offline a few times, switched weapons from the AR to the pistol, and ive prob jumped off the cliff on pillar of autumn over 300+ times

BUNGIE IM RUNNING OUT OF F****** ideas here...

tl;dr

Just keep trying. Took me a long time to get it, but I know people who got it almost immediately. A matter of luck, in a lot of ways.

Do it on Pillar of Autumn at the start of the level, it doesn't matter if anybody else is in the game (I got it alone, but I've been in-game when other people have gotten it). It works with either AR or pistol (I think pistol is easier, but either way), I had IWHBYD and Grunt Birthday Party skulls on, armor doesn't matter.

make sure u do it before you get too close to the elite or it wont work

i got it on like my 50th try, it worked when i was exactly above the elites head and then did the assassination
just gotta get lucky

its not glitchy
people who say that r just noobs
when u do the assassination but dont get the achievement u where to close to the ground

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You want to press the button when you're right behind the elite's head, with your legs behind his shoulders. If you get the assassination animation but don't get the achievement, it's because you weren't going to die from the fall (either you forgot to cancel your sprint or you landed on the elite's head, which lets you survive).