Choose an identity that matches the expression, keep equality on the common allowed domain, and retain every original restriction. One numerical check may catch a mistake; it does not prove an identity. Dividing an equation by a potentially zero factor can lose a solution.
What a strong response shows
Use to replace a matching squared pair.
Match the plus/minus signs in an angle-sum identity; do not distribute sine or cosine over addition.
When cancelling or dividing, record excluded inputs and check whether a zero-factor branch was lost.
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What You Need to Know
Equivalent trigonometric forms can expose a value, simplify an expression, or make an equation solvable.
Choose the identity that moves the expression toward the requested form.
Name the target form, apply one identity, and check that both sides still agree for allowed inputs.
Expanding every identity without a target.
What should guide identity choice?
Learning Objectives
3.12.A — Rewrite trigonometric expressions in equivalent forms with the Pythagorean identity. 3.12.B — Rewrite trigonometric expressions in equivalent forms with sine and cosine sum identities. 3.12.C — Solve equations using equivalent analytic representations of trigonometric functions.
Use Pythagorean or sum identities when they reduce the number of different trig forms.
. Identify the unwanted term. . Choose an identity containing it. . Substitute carefully. . Verify with a simple angle.
Using an identity with mismatched signs or arguments.
Which identity links sine and cosine squares?
Key Concepts
An identity is true for every input where both sides are defined; an equation is true only for selected inputs.
Simplifying an identity preserves all allowed inputs; solving an equation finds a set.
Keep equality transformations reversible and note any denominator restrictions.
Treating one numerical check as a proof of an identity.
What makes an identity different from an equation?
Relevant Representations
A symbolic identity can be checked with a unit-circle value, graph, or numerical table.
| Goal | Useful representation |
|---|---|
| prove equivalence | symbolic identity |
| check a value | unit circle or table |
| see solution count | graph |
| interpret angle | triangle or circle |
Use symbolic work for proof and a representation for a quick check.
Rewrite the expression, then test one allowed angle to catch sign or argument errors.
Testing an angle where a denominator is zero.
Which representation helps catch a sign error quickly?
Essential Relationships or Formulas
The Pythagorean and sum identities connect common trigonometric forms.
Match the identity to the exact structure, including parentheses and signs.
Rewrite one side, collect like terms only after the identity is substituted, and check restrictions.
Changing the sign in the cosine sum identity.
Which sum identity has a minus between the products?
Worked Examples
A targeted identity step is clearer than a long expansion.
Work from the more complicated side toward the simpler target.
Replace a squared trig term with its Pythagorean partner, simplify, and verify at an allowed angle.
Replacing a term in the wrong direction and making the expression longer.
Which side should you start with?
Common Mistakes
Identity errors are usually sign, exponent, and domain errors.
Check signs, squares, and denominators after every identity step.
Underline the exact part matched by the identity before substituting.
Reading a squared function as the square of the angle.
What does the exponent apply to?
AP Exam Tips
Write the identity before using it so the structure is visible.
Use the Questionbank set below to test this skill.
Keep arguments grouped and state restrictions when denominators appear.
Finish with a short equality chain and one allowed-value check.
Claiming an identity from one calculator output.
What evidence makes an identity argument general?
Can you transfer the method?
Rewrite using sine and cosine of , then evaluate it at .
Compare your reasoning
. At , the result is , agreeing with .
Work independently, then compare the mathematical evidence in your answer.