Completing a lost sector will unlock its Legend/Master version if:
It is the daily legend/master lost sector
You are over 1200 power

If these requirements are met you can access the legend/master lost sector via adventure flag near its entrance.

There are only two legend/master lost sectors at a time, one legend, and one master. They rotate daily. So far these have only appeared on Europa and the Cosmodrome. They may appear on other destinations in the future.

Completing a legend/master lost sector solo will grant a specific type of exotic armor, i.e. arms, helmet, chest, etc. The drop is not guaranteed. This is how you get your Beyond Light exotic armor, but you may get a different piece of exotic armor. For example: You may want the Icefall Mantle exotic titan gauntlets, but you might get an Aeon Safe instead.

On the Destiny 2 companion app the Beyond Light exotic armor states "Source: Legend and Master Lost Sector completions."

More info:

Legend difficulty is 1250 and Master is 1280. There are champions. You only get 3 revives, if you die after you run out of revives you have to load up the mission again. You have 15 minutes to complete the mission.

For everyone wondering, all new exotic weapons for BL are obtained via quest or activity completion, no world drops :/

And Claude, I bet that’s what the notification for a new Adventure was we talked about with Peter last night.

Some other information I’ve seen around is that each day is slot based, and it sounds like they are grindable
So if it’s Boots day and you want better rolled Gemini Jesters, you can run it a bunch to try for a drop.

I love this, I’ve been taking for weeks about wishing we could target specifics exotics somehow.