Destiny 2: How To Complete the Fourth Horseman Quest

How to complete the quest to acquire the Fourth Horseman exotic in Destiny 2. 

Step by Step Guide to the Fourth Horseman Quest

1. Head to the Tower.
2. There is now a walkway in the empty space between where Zavala and Shaxx stand.   Jump down to that walkway.
3. Follow the walkway and down the stairs as far as you can. 
4. You will see a door at the far end of the area.   It should be open.  If not, reload the Tower.  You are experiencing a known bug. 
5. Go through the door and head down until you see the vault. 
6. Behind the vault are more stairs.  Go down those stairs. 
7. It's parkour time.  Jump across the chasm and go as far as you can.  
8. When you seem to be at the end jump on top of the shelves and go into the ducts. 
9. Jump down into the vault and open the weapon box. 
10. That ends the Tower portion of the quest.  
11. Go see Ana
12. Grind Cabal. 

Here is a great video guide if you get stuck!

Datto has a complete walkthrough in this video.