After placing the two Ankhs in the receptacles on top of the sphinx, climb back onto the sphinx's headdress on the right side (when facing the rear of the sphinx). Turn right and walk out to the corner overlooking the rocky cave wall. Turn around so Lara is facing the top of the sphinx's head (as shown above).
Slide down the side of the headdress, holding Action to grab the edge. Drop to the ledge below, losing a little bit of health. Or, to avoid taking any damage, traverse to the right and drop onto the darker ledge. Climb down onto the sphinx's paw and slide down to the door you just opened.