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Defeat all bosses with only normal attacks.

This achievement is specifically for the bosses at the end of the stages. Not the mini bosses.
These are the bosses you need for the achievement:

City Stage boss (you have to beat this one with your mighty buster anyway)
Pryogen
Cryosphere
Dynatron
Seismic
Battallion
Brandish
Aviator
Countershade
Prison Stage boss (Call has no other weapons so this one is automatic)
Robot Factory Stage boss
Trinity (Trinity counts as two different bosses, but you have to beat both sections in one life for it to count as the achievement.)

The hardest part is Trinity. You can't use any of the weapons to avoid attacks. You will need a health tank in order to survive. I ended up getting the "I'll see you in Xel" achievement while attempting this fight. For the first part of the Trinity fight you need to memorize the order of the projectiles she shoots out to dodge them. While playing on normal the projectile pattern is the same every time so it just takes repetition.
For the second section most of the damage you do will come from when she is moving back and forth. Do not wait until she is just a head on the floor to damage her. Every time I destroyed a base I would jump and shoot her head as I dropped and then destroyed the next base. Beating her is slow going so you gotta be patient.

The rest of the bosses are easy. While not as damaging as the weapons the boss is weak to the mighty buster and you will be able to take them down.