For Our Princess! is a side quest in Tears of the Kingdom, which is part of a larger quest: Potential Princess Sightings! You'll need to complete this larger quest chain to get your hands on the Froggy Armor set!
As you're investigating all the stables in Hyrule for Traysi in search of clues about Princess Zelda's whereabouts, you'll come across a curious bunch of researchers at the Foothill Stable.
Updated by Hamza Haq on July 5th, 2023: The contents of the article have been adjusted to give be more accurate. We've also included some new links to point readers to relevant content that has been published recently.
Where To Start For Our Princess! Quest
Head over the Foothill stable by teleporting to the Upland Zorana Skyview Tower and paragliding north. You'll notice Penn near the stable, standing by an overhang. Talk to him about the researchers at the Stable and he will tell you that the researchers are somehow convinced that by the orders of the Princess, they need to survey the area and study the monsters therein wearing only underpants.
How To Complete For Our Princess! Quest
Penn will be perplexed and will ask Link to investigate. A new marker will appear at the entrance of Death Mountain, which is where you will need to head next.
As you reach the marker, you will see the mouth of a cavern and two men in their underpants hyping themselves up and trying to drum up the courage to investigate the mountain without any equipment.
Interrupt them, and you will overhear their strange conversation. Link will offer his help, saying he will head inside the cavern without any equipment to show them how it's done.
Now, it's time for action. Just like a combat shrine, you'll be stripped of all your weapons and armor and will be unable to save while you're inside the cave. Continue inward until you see a few Bokoblins and a Moblin standing around a fire.
Crouch so that they don't hear you and collect any weapons you see. There are no elixirs or armors to help you go unnoticed so be extra careful to not attract attention before you’re ready.
There are a few basic weapons, like a thick stick, a rusty sword, and a spear lying around. Pick them up as you circle the cave, all the while keeping yourself out of their line of sight.
There is one armored enemy in the group, so fuse a weapon with a boulder to easily destroy his armor when you engage. Initiate the fight by sneak attacking the Bokoblin at the far end of the cave.
Kill off all the enemies to make the Monster Forces Health Bar go down to zero. When you're done, collect anything valuable and head back to the entrance.
Talk to the two researchers, and they'll invite you back to the stable. Teleport over, and you'll see a large group of researchers expressing amazement at your impossible feat.
Initiate a conversation to find out that Zelda did not, in fact, say to explore in underpants; the workers just misunderstood. With that, the quest, For Our Princess! will be marked complete.