Go to the northwest corner of the cave, beyond the sphinx's right paw, to a sandy area with square pillars of various heights. High above on the right is a switch. That's where we're headed. Climb onto the lowest block.
Take a running jump to the next low block, up the slope near the cave wall. Pull up onto the ledge above.
Take a running jump to land on (or grab) the second tallest pillar.
Turn left and take a running jump (or a standing jump from the corner) to land on top of the tallest pillar.
Turn right, walk to the edge, and pull up onto the jutting ledge above.
(There's a save crystal here in New Game+.) Proceed forward carefully. To avoid the metal jaw trap just around the next corner...
...take a running jump over to the walkway on the right, so Lara lands near the small medi pack (6) Grab it, turn around...
...and follow the ledge toward the switch. Be careful to jump across the gap in the path just before the switch. Throw the switch to open a gold door in the southeast corner of the cave.
A winged mutant (4) should now fly in from the south (beyond the sphinx), but many remaster players have found that it gets stuck in the cavern wall, so you're unable to fight it until you get closer.
If the mutant does spawn correctly, I still recommend making a quick descent so you don't have to fight on the high ledge. Take a standing jump from the edge of the switch ledge down to the one with the large medi pack (7).
Turn around and hang from the far edge of that ledge (so Lara's back is toward the sphinx). Drop and slide down the sloping rocks to the ground, pressing and releasing Action as Lara slides...
...to break her fall.
If the winged mutant (4) did spawn correctly, it should approach from the south, though it may come around either the front or back of the sphinx. Shoot it from the ground as it flies toward you. Try to kill it before it touches down, so it doesn't physically attack Lara.
And keep your distance as you fight, since the explosion when it dies can injure her. (If the mutant is still stuck in the wall, don't worry. It will activate as we get closer, so you'll still get the kill.) Return to the front of the sphinx.
DARK MODE: Even with brightness adjusted, dark images on a light background can be hard to see. May I suggest a reputable dark mode browser extension, such as Dark Reader? (Download only from trusted sites, like darkreader.org or your browser's web store.)
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