Lesson 2: Proving Trigonometric Identities

Introduction to Trig Identity Verification

• Main goal: Prove one side of the trig equation equals the other
• Tips for verification:
  • Start with the more complicated side.
  • Convert expressions to sine or cosine.
  • Use algebraic operations such as:
    • Adding fractions
    • Factoring
    • Distributing
    • Squaring terms
    • Multiplying by conjugates
    • Clever forms of 1
  • Use trigonometric identities:
    • Pythagorean identities
    • Reciprocals (e.g., cosine and secant, sine and cosecant)
  • Always check the transformation path.
  • Multiple methods can prove an identity.
  • Conclude proofs with QED (Quod Erat Demonstrandum).

Example 1: Algebraic Identity

• Problem: Combine fractions with different denominators.
• Solution:
  • Factor numerators and eliminate denominators.
  • Simplify expressions to prove equality (e.g., 2 = 2).

Example 2: Trig Identity with Secant

• Problem: Work with secant and Pythagorean identities.
• Solution:
  • Rewrite cosine squared as secant squared.
  • Use secant squared minus 1 equals tangent squared.

Example 3: Tangent and Cotangent Conversion

• Convert tangent and cotangent using quotient property.
• Prove left side sum equals right side product.
• Solution:
  • Find common denominator.
  • Rewrite 1/sine as cosecant and 1/cosine as secant.
  • Prove by rearranging terms.

Example 4: Addition Problem

• Problem: Match expression to a given form.
• Solution:
  • Use common denominators.
  • Convert secant squared minus 1 to tangent squared.
  • Simplify to match target expression.

Example 5: Sine and Cosine

• Problem: Prove equation using sine and cosine.
• Solution:
  • Use Pythagorean identity: sine squared = 1 - cosine squared.
  • Simplify to confirm equality.

Example 6: Pythagorean Identity Creation

• Use identity 1 + sine.
• Simplify by distributing and factorizing.
• Prove left equals right.

Example 7: Complex Trigonometric Expression

• Rewrite and simplify complex fractions.
• Use identities to prove equality.
• Solution involves:
  • Factoring
  • Using Pythagorean identities
  • Simplifying to a common trigonometric form.

Conclusion

• Practice makes proving identities easier.
• Each problem is a puzzle requiring logical steps and verification.