AP PRECALCULUS — UNIT 3B · TRIGONOMETRIC & POLAR FUNCTIONS
3.12A — Equivalent Representations of Trigonometric Functions
Notes — Pythagorean and Reciprocal Identities
💡 Learning Objectives (3.12.A Part 1)
By the end of this lesson you will be able to:
- Use Pythagorean identities to rewrite trig expressions
- Convert between trig functions using reciprocal and quotient identities
- Verify simple trig identities by manipulating one side
- Simplify expressions by combining reciprocal, quotient, and Pythagorean identities
1. The Three Pythagorean Identities
💡 Pythagorean Identities
- sin²(θ) + cos²(θ) = 1
- tan²(θ) + 1 = sec²(θ)
- 1 + cot²(θ) = csc²(θ)
Each is just a rewriting of the unit-circle equation x² + y² = 1. The first one is the foundation; the other two come from dividing through by cos² or sin². Memorize all three — they will reappear constantly.
2. Reciprocal and Quotient Identities
💡 Six Identities to Memorize
- csc θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ = cos θ/sin θ
- tan θ = sin θ/cos θ
- sin θ · csc θ = 1
- cos θ · sec θ = 1
3. Strategies for Verifying Identities
To prove that two expressions are identically equal, work with ONE side at a time and transform it until it matches the other. Some go-to moves:
- Rewrite everything in terms of sine and cosine
- Use a Pythagorean identity to replace 1 − sin² with cos² or 1 − cos² with sin²
- Multiply by a clever form of 1 (like (1 − sin θ)/(1 − sin θ)) to simplify
- Combine fractions with a common denominator
- Factor when possible — including factoring sin² + cos² as (sin)(sin) + (cos)(cos), or recognizing differences of squares
⚠️ Important rule
When VERIFYING an identity, do NOT manipulate both sides at once. Pick one side (usually the more complicated one) and transform until you reach the other side. Otherwise you risk assuming what you're trying to prove.
4. Worked Example — Verifying
📘 Example — Verify (sin θ + cos θ)² = 1 + 2 sin θ cos θ
- Start with the LEFT side: (sin θ + cos θ)²
- Expand: sin²θ + 2 sin θ cos θ + cos²θ
- Apply Pythagorean identity sin²θ + cos²θ = 1: 1 + 2 sin θ cos θ
- Result matches the right side. ✓
📘 Example — Verify (1 − cos²θ)/(sin θ) = sin θ
- Start with LEFT: (1 − cos²θ) / sin θ
- Replace 1 − cos²θ with sin²θ: sin²θ / sin θ
- Cancel one sin θ: sin θ
- Result matches RIGHT. ✓
5. Simplifying Expressions
Use the same identities to reduce a complicated expression to a simpler equivalent — even when no proof is required.
📘 Example — Simplify
- (sec²θ − 1) / sec²θ
- Numerator: sec²θ − 1 = tan²θ (Pythagorean)
- Now: tan²θ / sec²θ = (sin²θ/cos²θ) / (1/cos²θ) = sin²θ
- Final: sin²θ
6. Common Identity Patterns
These ‘shortcuts’ come up so often that recognizing them saves time:
- 1 − cos²θ = sin²θ
- 1 − sin²θ = cos²θ
- sec²θ − 1 = tan²θ
- csc²θ − 1 = cot²θ
- (1 − cos θ)(1 + cos θ) = 1 − cos²θ = sin²θ
- tan θ · cos θ = sin θ
- cot θ · sin θ = cos θ
7. Summary
- Three Pythagorean identities, plus six reciprocal/quotient identities — memorize them all
- To verify, transform ONE side until it matches the other
- First-aid moves: convert everything to sin and cos, use a common denominator, factor, apply Pythagoras
- Recognize quick-substitution patterns to skip routine algebra