Go around to the far side of the sphinx and climb up onto its left paw (i.e., the one on the right when facing it). Climb the ledges on its shoulder to reach its back.
Step up onto the low angled block on the sphinx's back. From there, take a running jump to grab the ledge on the back of its head.
Hold Walk while hanging from this ledge, then press Forward to pull up into a handstand walkover. The "Pet This Cat" achievement/trophy should now pop.
NOTE: The official text says, "Do a handstand on the Sphinxes face," and it definitely works here (above) at the front of the sphinx, but the back of the head is also fine.
Climb to the top of the head. Place one of the ANKHS in the receptacle here.
Go around either side of the head, hop down onto the ledge above the sphinx's forehead, and put the second ANKH in its matching receptacle. This opens the door between the sphinx's paws. (NG+ players also find a save crystal on this ledge.)
NOTE: If you haven't already gotten SECRET #1, the UZIS and accompanying ammunition (12-13), you can do that now, as described in the footnote below the main walkthrough.
To descend quickly and safely, first climb back on top of the sphinx's head, on the right side when facing the rear of the sphinx. Facing the ledge in front of the first Ankh receptacle. Hold Action and hop back so Lara slides down the side of the head...
...and grabs the edge. Let go to land on the sphinx's shoulder. Then hop down onto the "collarbone" ledge, then the ledge below...
...then the sphinx's paw, and finally the ground near the door you just opened. If you didn't already get the magnum clips (5) on the ground here, pick them up now.
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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