Chapter 43: Trigonometric Identities: Situation 37 From the Macmtl-Cptm Situations Project

While proving a trigonometric identity a student produced the following sequence of equations.

When asked about her reasoning, the student replied, “I just treated the equation like any algebra equation. You know, what you do to one side, you have to do to the other, and then I showed it was the same as 1 = 1. I know 1 = 1 is true, so the identity must be true.”

Proof in trigonometry is like proof in other areas of mathematics: One cannot assume a result in order to prove it but must progress by logical steps from a given statement or established result to the conclusion. When reasoning using a series of equations, it is important to show that each step necessarily and sufficiently follows from the previous step. The facts that only some manipulations of equations are reversible and that not all algebraic operations have inverses that preserve the domain of the original variable can introduce logical errors in trigonometric proofs.