Preheat the oven to 375 degrees F., and line a baking sheet with parchment or a baking mat.
In a medium bowl, cream together the butter, shortening, and the sugar until it is light and fluffy, about 3 minutes.
Add the eggs and vanilla, and mix until well combined.
Add the flour, cream of tartar, baking soda, and salt, and mix well to combine.
In a small bowl, mix together the remaining sugar and cinnamon.
Working with a large tablespoon of dough at a time, roll the ball between your hands (it might be a little sticky, so grease your hands a bit if you need), and then roll the dough in the cinnamon-sugar mixture, coating the ball completely.
Place the coated balls on the prepared cookie sheet 2 inches apart, and repeat with remaining dough.
Bake for 8 to 9 minutes. Let the cookies rest on the cookie sheet for 5 minutes after you remove them from the oven, and then remove them from the pan to let them cool on a wire rack.
Serve warm, and store any leftovers store in an airtight container.
Notes
I tried and tried to make a snickerdoodle cookie recipe with only butter, no shortening, and I could not do it. I'm convinced that shortening is okay in a few circumstances, and this is one of them.