The following page contains the description for completing the Trials of the Jaguar and the Spider in Shadow of the Tomb Raider (the Path to the Hidden City mission). The Trial of Jaguar will require Lara to swim across the flooded part of the ruins, and in the Trial of the Spider you'll have to complete a short climb. Luckily, both trials are neither long nor difficult.
The Trial of the Jaguar
Begin with jumping down the destroyed stairs and into the water. Dive and swim towards the main tunnel. At one point Lara will have to squeeze through a very narrow gap. Don't waste any time, so you don't run out of air. After you're through, you will be attacked by an eel. Mash the attack button, to take out the fish with the knife. Then, quickly head for the next air pocket.
Proceed with the exploration. Lara's objective is the narrow passage between the rocks. Squeeze through and swim to the surface.
The Trial of the Spider
The new portion of the ruins constitutes the Trial of the Spider. Approach the wall with the red silhouette and climb the higher shelf to begin the main climb. You'll need to use the axe wall and reach the upper level. Now squeeze through the narrow corridor.
Picture 1 shown the spot where you'll need to use the grappling axe (the character has it). Perform a long jump and swing it in the air by pressing the climbing axe button. You'll catch a spot on the ceiling. Swing on the rope and jump to reach the hole above the red silhouette (picture 2). The corridor will lead you to a separate location, where you'll face the Trial of the Eagle.