Lesson 3: Reciprocal, Quotient, and Pythagorean Identities
Estimated time: 30-35 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- State all reciprocal, quotient, and Pythagorean identities from memory
- Use identities to simplify trig expressions
- Derive the alternate Pythagorean identities (1 + tan² = sec² and 1 + cot² = csc²)
- Write one trig function in terms of another using identities
Reciprocal Identities
csc θ = 1/sin θ sec θ = 1/cos θ cot θ = 1/tan θ
Equivalently: sin θ = 1/csc θ, cos θ = 1/sec θ, tan θ = 1/cot θ
Quotient Identities
tan θ = sin θ / cos θ cot θ = cos θ / sin θ
Example 1: Simplifying with Quotient Identity
Simplify: sin θ / cos θ · cos θ
Solution: (sin θ / cos θ) · cos θ = sin θ. The cos θ cancels.
The Three Pythagorean Identities
Identity 1: sin²θ + cos²θ = 1
Identity 2: 1 + tan²θ = sec²θ
Identity 3: 1 + cot²θ = csc²θ
Identity 2 is obtained by dividing Identity 1 by cos²θ:
sin²θ/cos²θ + cos²θ/cos²θ = 1/cos²θ ⇒ tan²θ + 1 = sec²θ
Identity 3 is obtained by dividing Identity 1 by sin²θ:
sin²θ/sin²θ + cos²θ/sin²θ = 1/sin²θ ⇒ 1 + cot²θ = csc²θ
Example 2: Using a Pythagorean Identity
Simplify: sec²θ − tan²θ
Solution: From Identity 2: sec²θ − tan²θ = 1
Example 3: Writing in Terms of Sine and Cosine
Simplify: (1 − cos²θ) / sin θ
Solution: 1 − cos²θ = sin²θ (from Identity 1). So sin²θ / sin θ = sin θ
Example 4: Factoring with Identities
Simplify: sin²θ − 1
Solution: sin²θ − 1 = −(1 − sin²θ) = −cos²θ
Writing One Function in Terms of Another
Example 5: Express tan in Terms of sin
Write tan θ in terms of sin θ only (assume θ is in QI).
Solution: tan θ = sin θ / cos θ. Since cos θ = √(1 − sin²θ) in QI:
tan θ = sin θ / √(1 − sin²θ)
Check Your Understanding
1. Simplify: csc²θ − cot²θ
2. Simplify: (sin²θ)(1 + cot²θ)
3. Simplify: tan θ cos θ
Key Takeaways
- There are three reciprocal, two quotient, and three Pythagorean identities.
- The Pythagorean identities are all derived from sin² + cos² = 1.
- To simplify, try converting everything to sin and cos, then cancel or apply identities.
- These identities are used constantly in later chapters for verifying and solving equations.