These figures show proofs of trigonometric addition and subtraction formulas for cosine and sine by Leonard M. Smiley, University of Alaska Anchorage. The subtraction formulas appeared in the Month, Year issue of Mathematics Magazine.

The proofs are valid for "small" angles only, but are completely *synthetic* and use a common, minimal diagram from Euclidean geometry. The claim of minimality is supported by the absence of any unused points and the low number (5 each) of points and segments. The proofs share nothing with the standard *analytic* proofs using Cartesian coordinates, and thus are fully complementary to these more general treatments.

(Notes: Screen resolution of 1024 X 768 or higher is recommended for each proof to be fully visible on one screen with no scrolling. Diagrams produced by Deborah Smiley of Ivyworks)

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Len Smiley Deborah Smiley