A unified geometrical approach for trigonometric angle sum and difference identities

Two important pedagogical techniques for developing deeper mathematical understanding are to prove a given theorem in different ways and to unify the proofs of different theorems. Trigonometric angle sum and difference identities are introduced in Unit 2 of Specialist Mathematics in the Australian Curriculum (Australian Curriculum, Assessment and Reporting Authority, n.d.). Typically, students first see these identities derived using an admixture of geometrical and algebraic methods, as exemplified in Weisstein (2018). In later (generally early tertiary) mathematics courses, complex exponential forms of the trigonometric functions and the Euler formula provide a more consistently algebraic approach to these identities. On the other hand, several geometric constructions are available for the various cases (Kung, 2008; Ren, 1999; Smiley, 1999; Smiley and Smiley, 2018) while Nelsen (2000) provides an excellent although apparently not widely known figure from which all six trigonometric angle sum and difference identities (for acute angles) may be derived geometrically. In this paper an alternative, unified and simplified geometrical development of these identities is presented based on what is also in essence a single geometric construction. Allowing students already familiar with the traditional derivations of the trigonometric angle sum and difference identities to work through this alternative approach provides opportunities to deepen and consolidate their understanding of both the identities themselves and the geometrical techniques involved.