Lesson 1: Strategies for Verifying Trig Identities
Estimated time: 35-40 minutes
Learning Objectives
- Distinguish between an equation and an identity
- Apply systematic strategies to verify identities
- Convert expressions to sine and cosine
- Factor and combine fractions in trig expressions
What Is a Trig Identity?
Trig Identity — An equation involving trig functions that is true for all values where both sides are defined. Unlike equations (solve for θ), identities are verified by transforming one side to match the other.
Golden rule: Work on one side only. Transform the more complex side to match the simpler side.
Key Strategies
- Convert to sin and cos: Replace all trig functions with sin and cos.
- Factor: Look for common factors or difference-of-squares patterns.
- Combine fractions: Use a common denominator.
- Multiply by conjugate: Useful when 1 ± sin or 1 ± cos appears.
- Use Pythagorean identities: sin² + cos² = 1 and variations.
Worked Examples
Example 1: Convert to Sin and Cos
Verify: tan θ cos θ = sin θ
LHS: tan θ cos θ = (sin θ/cos θ) · cos θ = sin θ = RHS
Example 2: Using Pythagorean Identity
Verify: (1 − cos²θ) csc²θ = 1
LHS: sin²θ · (1/sin²θ) = 1 = RHS
Example 3: Factoring
Verify: sin²θ − cos²θ = 2 sin²θ − 1
LHS: sin²θ − cos²θ = sin²θ − (1 − sin²θ) = 2 sin²θ − 1 = RHS
Example 4: Combining Fractions
Verify: 1/(1 − sin θ) + 1/(1 + sin θ) = 2 sec²θ
LHS: [(1 + sin θ) + (1 − sin θ)] / [(1 − sin θ)(1 + sin θ)] = 2 / (1 − sin²θ) = 2/cos²θ = 2 sec²θ = RHS
Check Your Understanding
1. Verify: cot θ sin θ = cos θ
2. Verify: sec²θ − 1 = tan²θ
3. Verify: (sin θ + cos θ)² = 1 + 2 sin θ cos θ
Key Takeaways
- Work one side only — never move terms across the equals sign.
- Convert to sin/cos when stuck.
- Look for Pythagorean substitutions and factoring opportunities.
- Multiply by conjugates when you see 1 ± sin or 1 ± cos.