Preheat the oven to 400°F. Lightly grease (or line with parchment) two baking sheets., In a medium-sized mixing bowl, beat together the shortening and sugar until smooth, then beat in the eggs, again beating until smooth. The mixture will become lighter, and lighter-colored, as you beat; this is the result of air being absorbed., Beat in the vanilla, cream of tartar, baking soda (or baking powder, in place of the cream of tartar and soda), and salt, then add the flour, mixing slowly until combined., Place about 1/2 cup cinnamon sugar in an 8" or 9" round cake pan., Drop the soft dough by tablespoonfuls into the pan with the sugar, about 6 to 8 balls at a time. Gently shake the pan to coat the dough balls with sugar., Place the cookies on the prepared baking sheets, leaving about 1 1/2" between them. Using the bottom of a glass, flatten each cookie until it's about 1/2" thick. Repeat until you've used up all the dough., Bake the snickerdoodles for 11 to 12 minutes, reversing the position of the pans (top to bottom, and back to front) midway through., Remove the cookies from the oven once they're set and just starting to turn golden. Transfer them to a rack to cool completely., Store cookies, tightly wrapped, at room temperature for several days; freeze for longer storage.