This achievement can be unlocked on any difficulty, and can be unlocked using NG+
In order to unlock this achievement you need to own each weapon in the game at one time.
*Please note that DLC weapons do not count towards this achievement*
Since you can only carry 4 weapons, you will need to store 4 more in your safe. You get the plasma cutter for free as part of the story during chapter 1, the other 7 weapons you will need to either find the schematic on the map, download it at a store, and then purchase the weapon or just purchase the weapon if it is available in the store.
To purchase all the weapons for the achievement you will need to have 55,000 credits, the cost breakdown is as follows:
Plasma Cutter: Free (story related)
Line Gun: 9,000
Pulse Rifle: 7,000
Flame Thrower: 11,000
Force Gun: 11,000
Ripper: 8,000
Contact Beam: 9,000
Total: 55,000
I have included a video to help clear up any additional confusion.
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