You have to either find a brown sheep or Dye it with cocoa beans, which you can find them in chests or in jungles
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There is no 'brown dye' - to dye wool brown, you just place it along with Cocoa Beans into your crafting slot, to make Brown Wool.
You can dye wool with it, or you can create decorative blocks of it using 9 pieces.
No, you need wool.
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You have to put a cactus in a furnace and then apply the resulting dye to the wool.