Trigonometric Identities and Simplification

Identity Simplification and Verification

Fundamental Identities Used for Simplification

• To simplify expressions, fundamental trigonometric identities will be employed.

Example Problems

### Problem 1: Simplifying Cosine Expression

• Given:

  • ext{cos}^2 x - 9
  • ext{COS} x - 3
  • First factor as follows:
  • Results in:
    • ext{Cas}x - 9
    • ext{COS} x imes 3

  Problem 2: Factor the expression

• Factor:

  • 5 ext{cos}^2 x + 9 ext{cos} x - 2
  • Results in:
    • (5 ext{cos} x + a ext{cos} x - 2)

  Problem 3: Basic Trigonometric Identities

• Apply identities to simplify:

  • ext{sin}(90° - x)
  • ext{sin}(-x)
  • Results in:
    • ext{Sin}(go - x)
    • ext{sin}(-x)

  Problem 4: Difference between Squares

• Given:

  • ext{sin}^2 x - ext{cos}^2 x
  • Factor as:
    • ext{sin} x - ext{cos} x

Verification of Identities

• You must show all work clearly for credit.

Identity Verification 1

• Verify the identity:
  • Statement:
    • ext{csc}^2 heta - ext{cot}^2 heta = ext{cos}^2 heta
  • Rearranging gives:
    • ext{tan}^2 heta ext{csc}^2 heta
    • Hence yields the identity:
    • ext{csc}^2 heta - ext{cot}^2 heta = ext{cos}^2 heta

Identity Verification 2

• Statement to verify:
  • 1 + ext{sec} x - 1
  • Transforming gives:
    • rac{1}{ ext{sec} x + 1} = 2 ext{cot} x ext{csc} x
    • Showing:
    • 1 ( ext{sec} x + 1) + 1( ext{sec} x - 1) = 2 ext{cot} x ext{csc} x

Identity Verification 3

• Given:
  • ext{sec}^2(2t) - 1
• Applying:
  • Pythagorean identity results as a difference of squares.

Identity Verification 4

• To verify:
  • ext{(csc}^2 t - 1)( ext{cos}^2 t - 1) = - ext{cos}^2 t
  • This is a rearranged form and needs simplification for verification.

Identity Verification 5

• Verify:
  • ext{sec} x - ext{cos} x = ext{sin} x ext{tan} x
  • Which leads to:
    • rac{ ext{sec} x}{ ext{Cos} x} = ext{sin} x ext{tan} x
    • Complete verification method and approaches must just demonstrate clear mapping to established identities.