Direct geometical proofs of the addition theorem, and others allied to it, are often valid for only a limited range of values, and the constructions used are applicable to only one set of formulaæ. In the following paper a method is given which is valid for angles of all values and which is applicable to all the usual formulae involving sines, cosines, and their simple products. One of each set is proved: the others may be obtained either directly or by substitution in the usual manner.