There is no way.
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I think you are mistaking the orange color for brown. The orange do really look like brown on normal capes. Anyways, mix red dye with yellow dye for orange dye. When you use the orange dye on a normal cape, you will get your " brown" cape.
mix red and blue dye Red dye- redberries blue dye- woad leaves
You get red dye by visiting the witch in draynor (location to the west of lumbridge) and giving her 5gp and 2 redberries. She then gives you red dye in exchange.
Orange dye can be made by putting yellow dye (from crafting yellow flowers) and red dye (from crafting red flowers) on the crafting table.
Red #40 is NOT made from beetles. You are thinking of another red dye. Red 40 is made from petroleum.