Me and my son are playing the very first tomb raider game. I had it when..

After the three gates are closed and the boulder falls down, if you look into the pit from the front you can see where the small wooden platform is. This can be moved, so climb down and move the first column to one of the light spots on the floor there, then move the wooden thing over by the wall where the ledge is and jump up and climb to the walkway, then use the center platform to jump onto the column you just moved to the light spot.

From the column jump to the right-hand ledge and work your way around to jump to the ledge to the right, climb the statue and then climb and jump onto the small roof, turning around to face the pit. See that gong? Shoot it and you will cause some of the center area to crumble and fall, at which point you can now use the grapple to swing across where you will find a lever you can pull, opening the first gate.

Now you just need to work your way through the right passage getting past all the traps to eventually reach the next lever, pull that, then make your way to the central chamber where the third lever is, and pulling that opens the last gate so you can continue the story.