Here is an interesting comparison of Cortana over the course of the 4 halo games:
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Spawning in Floodgate, any difficulty, follow the path in town, once you get to the gates that are damaged, go into the left alley, and go into the courtyard with Marines and Flood fighting, go into the structure on the right of the ramp and see a Marine get infected, dodge him and go left into the big Marine and Flood battle go into the next big room and get onto the bridge in the middle by taking some stairs near the right of itself. Go across and into the room with a hole in the middle of it, jump down and a cutscene of Cortana is there, get out of the room by walking through some doors in the back, this will take you on the other side of where the ramp was in the courtyard. On the right wall of the room, there should be a door that leads to a small room, and some more Flood will come after you. And there will be 2 walls splitting the room, ignore the doors, you can't go through them. Once your done moving around the second wall, there should be an open door, and by now you can hear the Marine sobbing about killing his infected teammates. If you stay there long enough (I estimate 10 minutes or so, maybe less.) the Marine that was sobbing will shoot himself in the head with a pistol. You can kill him also.