Algebra 101 - Unit 15

Lesson 95: Trigonometric Identities (Pythagorean, Reciprocal, etc.)

Trigonometric identities are equations involving trigonometric functions that are true for all valid values of the variable. They simplify expressions, solve equations, and prove other identities.

Basic Identities

• Reciprocal Identities:
  • csc θ = 1 / sin θ
  • sec θ = 1 / cos θ
  • cot θ = 1 / tan θ
• Quotient Identities:
  • tan θ = sin θ / cos θ
  • cot θ = cos θ / sin θ
• Pythagorean Identities:
  • sin² θ + cos² θ = 1
  • 1 + tan² θ = sec² θ
  • 1 + cot² θ = csc² θ

Examples

Example 1 — Using Reciprocal Identities: Simplify csc θ · sin θ → csc θ · sin θ = 1

Example 2 — Using Quotient Identity: Simplify tan θ / (sin θ) → (sin θ / cos θ) / sin θ = 1 / cos θ = sec θ

Example 3 — Using Pythagorean Identity: Simplify 1 − cos² θ → 1 − cos² θ = sin² θ

Practice Problems

1. Simplify sec θ · cos θ.
2. Simplify cot θ · tan θ.
3. Prove 1 − sin² θ = cos² θ.
4. Simplify (1 + tan² θ) / sec² θ.
5. Express cot θ in terms of csc θ and sin θ.