Machin's Formula

$${\textstyle{1\over 4}}\pi = 4\tan^{-1}({\textstyle{1\over 5}})-\tan^{-1}({\textstyle{1\over 239}}).$$

There are a whole class of Machin-Like Formulas with various numbers of terms (although only four such formulas with only two terms). The properties of these formulas are intimately connected with Cotangent identities.

See also 196-Algorithm, Gregory Number, Machin-Like Formulas, Pi