Character: "Solid Snake" / "Old Snake" / David
Chapter 2: Solid Sun Part 4

Laughing Octopus (1st phase) boss fight is more of just a stealth game against Snake. Use Solid Eye Night vision mode to find her, then pump her with bullets, and at the same time watch her movements. Whenever she is ready to fire, take cover and let her fire till she ran out of bullets before you return fire at her.
There will be one moment where she will disguise herself as Otacon's Mk. II. Take caution as I have written it is her in disguise, trying to lure Snake to her traps.

After Laughing Octopus is defeated, she will turn into Laughing Beauty (2nd phase). Throughout the whole MGS4, all the Beauty & the Beast bosses have 2 phases. Their Beauty counterpart is just a simple move of tossing stun grenades at them till their meter reach 0. That is all, but do aim carefully such that the tossed stunned grenade is in their line of sight. Otherwise it will be in vain.

Playlist: https://www.dailymotion.com/playlist/x73vz3