Preheat the oven to 400 degrees F (200 degrees C).
Mix together 1 1/2 cups white sugar, shortening, and eggs with an electric mixer in a bowl until creamy.
Sift together all-purpose flour, cocoa power, cream of tartar, baking soda, and salt in a separate bowl. Fold this mixture into the egg mixture until well blended.
Mix together 1/4 cup white sugar and cinnamon in a shallow bowl or on a plate.
Roll cookie dough into 1-inch balls; cover each in sugar and cinnamon mixture. Place on an ungreased baking sheet.
Bake in the preheated oven until soft and golden on top, 6 to 8 minutes, making sure they do not burn. Let cool before transferring to a wire rack. The tops will crack as they cool.
It's important not to over-bake the snickerdoodles as they will harden as they cool. They taste much better when they are a little chewy.