You don't have to. After you get the fur from the fence in Puerto Rico and test it, you return there and the farmer is already setting a trap. All you need to do is use the Jeep to corral 3 spotted goats into the clearing at Snaggletooth Rock (center star on the map). Then get out and join him.
Not really
Yes, you can find proof of The Jersey Devil, Bigfoot, Nessie, and the Chupacabra. (see related question)
The only trap on the island is provided by the goat herder at Puerto Rico. You can help him try to trap the chupacabra. On your first trip to Puerto Rico, get the fur from the chicken farm fence (right side star). When you analyze the DNA, it is not from a chupacabra. Go back to Puerto Rico, where the herder is preparing the trap. His brother suggests that you use the jeep to herd three spotted goats into the grass at Snaggletooth Rock (center star). The chupacabra will be lured there by the goats. Drive there, get the three goats into the rock, and then get out and see the herder.
On your second trip to Puerto Rico, the farmer will be trying to trap the chupacabra at Snaggletooth Rock (center star on the map). Drive there and use the jeep to corral 3 spotted goats into the area near the rock. Get out and meet the farmer, and soon you will capture the chupacabra (sort of).
You can't. But its tooth is proof enough that it is an unknown species.
On your second trip to Puerto Rico (the first time you get the fur), drive the jeep to herd 3 spotted goats into the place the chupacabra was spotted, the rocks at the center star. Then get out and meet the farmer, who has prepared a trap.
Take it back to Mews. Once you have gotten the photo of the Yeti footprint, and proved the other three cryptids (Nessie, chupacabra, and the Jersey Devil), Mews will send you to track down Bigfoot.
You can summon Nessie with the pennywhistle. You can lure the chupacabra (on your second visit to Puerto Rico) by herding 3 spotted goats into the area of Snaggletooth Rock (center star on the map). The farmer will prepare a trap there.
You have to look for Bigfoot/Yeti in the Himalayas, the Jersey Devil in New Jersey, the Loch Ness Monster in the Loch Ness in Scotland and the Chupacabra in Puerto Rico.
You need a photo of Nessie at Loch Ness, a tooth from the Chupacabra from Puerto Rico, and egg shells from the nest of the Jersey Devil in New Jersey. (see related solution)
On your first trip to Puerto Rico, you get some fur that turns out to be a wild dog's fur. On your second visit, the herder comes up with a plan to trap it at Snaggletooth Rock (center star). You have to use the jeep to herd 3 spotted goats into the spot in front of the cave. You get out and you'll meet the herder there. He has already set a trap and you wait behind the rocks. You will catch the chupacabra in this large crate. Things don't work out as planned, but you do get proof of the chupacabra to take to Mews.
On your first trip to Puerto Rico, you get some fur that turns out to be a wild dog's fur. On your second visit, the herder comes up with a plan to trap it at Snaggletooth Rock (center star). You have to use the jeep to herd 3 spotted goats into the spot in front of the cave. You get out and you'll meet the herder there. He has already set a trap and you wait behind the rocks. You will catch the chupacabra in this large crate. Things don't work out as planned, but you do get proof of the chupacabra to take to Mews.