Destroy The Destroyer

This is the mission to kill The Destroyer, an immensely powerful and dangerous being locked inside the Vault until Steele unknowingly releases it.

Objectives
Defeat the Destroyer.
 * Destroyer defeated: 0/1
 * "Do you feel that I betrayed you? Do not. I am possessed of a sight that allows me to look forward and backward along the timeline, and the release of this creature was inevitable. There was only one chance for this dimension's continued existence. A hero. Someone who was at the right place, at the right time, and strong enough to push back. I have given you the unique opportunity to defend the existence of all that you know. Go! Fight! Be that hero! I will help you however I can."

Preparation
This fight can be one of the hardest in the game if the appropriate preparations are not made. It is advisable to have at least one powerful, reasonably accurate weapon for each weapon class (Rocket Launchers are not preferable, as it can be more difficult to score critical hits on the eye). Having some proficiency in each weapon class helps as well.

The next issue is ammunition. The Destroyer can take hundreds of critical hits before dying, so ensure that you have plenty of ammo. Ammo Regeneration weapons or class mods work well when combined with lower-level SDU's.

An alternative would be to purchase Purple or higher SDU's for several weapon classes and to rely on the initial stockpile. This method is much riskier, because fighting through the Eridian Promontory without using ammo-regen can leave you facing The Destroyer without enough supplies to make the kill.

Eridian Weapons can be used, but due to the slow recharge times and accuracy problems (i.e. the wave blaster) it is a less appealing option.

Killing The Destroyer
Immediately take cover behind the short pillar to the left of where the character starts the battle (this pillar has a gray crate in front of it). You will remain at this location for the rest of the battle. If you are standing out in the open, The Destroyer will slam its red tentacles against the ground causing a shockwave that will throw the character back, or it will lash out with its tongue to the same effect. Your first priority is to destroy the purple tentacles that sprout from The Destroyer's back. These launch large purple needles that explode upon hitting the ground, dealing significant damage. Unfortunately, these needles often arc over the available cover, so you cannot simply ignore them. Accurate weapons of any type work well here. Shoot the glowing purple orbs near the middle of the tentacles to destroy them faster and get critical hits.

Once those are out of the way, equip a hard-hitting weapon with a high rate of fire and good accuracy (a scope also helps). Open fire on the glowing blue opening (Eye) just above where the Destroyer's mouth is located. If you score a direct hit on the eye, it will count as a critical. Keep ducking behind cover to dodge shockwaves and tongue attacks.

Another strategy that isn't used a lot, but is really effective, is: when the battle starts, just stay put and keep shooting the mouth. Don't shoot the tentacles yet because when you're downed you can use them to get easy second winds. Just keep doing this until The Destroyer is dead. Addendum: This works especially well with a double anarchy with 4 bullets per shot as this does some of the best damage per second in the game though it consumes an excessive amount of bullets and is relatively innacurate, the size of the Destroyer makes it very effective and it is possible to defeat it without ever being attacked.

There's yet another strategy which makes this fight trivial. As soon as the battle starts, go and run behind the top right pillar and look towards the right. Provided that your shield is strong enough or have transfusion greandes etc., you can shoot the visible tentacle without ever being hit by the eye beam or the shock wave. Missiles will still hit you, but your shield can easily absorb that much damage. This strategy makes the fight only a matter of time and you don't have to worry about ammo if you have Eridian weapons.

Also, if none of the above work for you, you can defeat him easily by exploiting game mechanics. Namely, by jumping off the cliff at the back of the area where you fight him. This will unfortunately cost you lots of money, seeing as you will respawn 20+ times, costing you money each time. However, when you suicide (Or get thrown off the edge by his shockwave) you will respawn with full health and shields, and his health will be in the same place as it was when you fell. Just unload as much ammo into him as you can while walking around to the cliff edge. Another helpful hint: If you reload right when you jump, (Even if the weapon isn't empty yet) you will finish reloading while the death scene shows, giving you a full clip to fire as soon as you come back. (Note: Only tested on unpatched Xbox 360 version. Need confirmation on other platforms.)

The Fight Continues
Once roughly 1/3 of The Destroyer's health is gone, it will edge out of the Vault a bit more and latch on to two pillars on the side of the arena (Note: it is possible to hide behind the Destroyer's right tentacle and simply melee it for the last third of its health (effectiveness of this has been disputed)). It will also sprout four more purple tentacles. These are a concern, but so is The Destroyer’s newest attack: the eye-beam. Take out the purple tentacles and watch out for an aura around The Destroyer’s eye. Once you see it, take cover behind the pillar. A thin blue beam will be fired for several seconds. Though it may appear as though the beam is passing directly through the pillar, it will not damage the character. The beam sound may continue after the beam has already left. To be safe, pop out for a second and look directly at the eye. If the beam is gone, open fire. Another tactic in avoiding the beam is that when you hear or see the beam about to be fired quickly run over to the fast travel station and enter it and wait for a short period of time. If done correctly the next beam it fires will not actually fire a beam therefore you will have extra time to gun it down. Watch out for purple tentacles, which may occasionally be re-grown. Repeat this process until The Destroyer is defeated.
 * Alternately, once you get the health down enough for him to grab the outermost pillars you can get behind them where rockets and his beam can't shoot and either shoot or melee him to death with no threat what so ever.

Video Walkthrough
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Simple Alternative Method
Despite being a boss battle, you can alternatively find ways to level up beyond its level. The effectiveness of your attacks will increase as normal, as well as the reduction in effectiveness of it's attacks. It is possible to engage the Destroyer at level 36 (with the Destroyer being level 33), without taking damage beyond shields if well prepared. Weapons recommendations remain as above.

All you need to do beforehand is to acquire as much experience through simple killing, exploration, achievements, and available missions.

Completion

 * "You did it! You won. I'm sorry I couldn't warn you, but for you to be there, it couldn't have unfolded any other way. You were brilliant, and your actions will not go unrewarded."