Valid identity proofs and counterexamples

Proving trigonometric identities and finding counterexamples

Learn to assess mathematical arguments, not just recognize a familiar formula. This precalculus practice set develops identity verification through justified algebra, domain checks, reversible transformations and carefully chosen counterexamples. Questions ask you to identify a missing reason, compare proof strategies and repair a flawed conclusion.

Single-select and multiselect tasks progress from the meaning of an identity to arguments involving squared expressions and cancelled factors. Explanations distinguish a proof for every allowed input from agreement at a few sample angles. Working on one side is useful, but valid two-sided equivalences can also establish an identity.

How to use this set

Write the domain and explain each equality before inspecting the options. For a false claim, seek one allowed input where the sides differ. For a proposed proof, check whether a step assumes the conclusion or loses sign information. Review fundamental identities and valid equation transformations when needed. These recognition questions support proof practice; writing a complete argument independently remains a separate learning task.