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:

    • extcos2x−9ext{cos}^2 x - 9
    • extCOSx−3ext{COS} x - 3
    • First factor as follows:
    • Results in:
      • extCasx−9ext{Cas}x - 9
      • extCOSximes3ext{COS} x imes 3
    Problem 2: Factor the expression

  • Factor:

    • 5extcos2x+9extcosx−25 ext{cos}^2 x + 9 ext{cos} x - 2
    • Results in:
      • (5extcosx+aextcosx−2)(5 ext{cos} x + a ext{cos} x - 2)
    Problem 3: Basic Trigonometric Identities

  • Apply identities to simplify:

    • extsin(90°−x)ext{sin}(90° - x)
    • extsin(−x)ext{sin}(-x)
    • Results in:
      • extSin(go−x)ext{Sin}(go - x)
      • extsin(−x)ext{sin}(-x)
    Problem 4: Difference between Squares

  • Given:

    • extsin2x−extcos2xext{sin}^2 x - ext{cos}^2 x
    • Factor as:
      • extsinx−extcosxext{sin} x - ext{cos} x

Verification of Identities

  • You must show all work clearly for credit.
Identity Verification 1
  • Verify the identity:
    • Statement:
      • extcsc2heta−extcot2heta=extcos2hetaext{csc}^2 heta - ext{cot}^2 heta = ext{cos}^2 heta
    • Rearranging gives:
      • exttan2hetaextcsc2hetaext{tan}^2 heta ext{csc}^2 heta
      • Hence yields the identity:
      • extcsc2heta−extcot2heta=extcos2hetaext{csc}^2 heta - ext{cot}^2 heta = ext{cos}^2 heta
Identity Verification 2
  • Statement to verify:
    • 1+extsecx−11 + ext{sec} x - 1
    • Transforming gives:
      • 1extsecx+1=2extcotxextcscx\frac{1}{ ext{sec} x + 1} = 2 ext{cot} x ext{csc} x
      • Showing:
      • 1(extsecx+1)+1(extsecx−1)=2extcotxextcscx1 ( ext{sec} x + 1) + 1( ext{sec} x - 1) = 2 ext{cot} x ext{csc} x
Identity Verification 3
  • Given:
    • extsec2(2t)−1ext{sec}^2(2t) - 1
  • Applying:
    • Pythagorean identity results as a difference of squares.
Identity Verification 4
  • To verify:
    • ext(csc2t−1)(extcos2t−1)=−extcos2text{(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:
    • extsecx−extcosx=extsinxexttanxext{sec} x - ext{cos} x = ext{sin} x ext{tan} x
    • Which leads to:
      • extsecxextCosx=extsinxexttanx\frac{ 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.