SECT 5.4 PROVINGTRIGONOMETRIC IDENTITIES
PROVING IDENTITIES ALGEBRAICALLY:
When proving trigonometric identities:
Convert all trig. functions to sine or cosine
Use basic trig. Identities to simplify complicated ones
Odd/Even, Quotient, Pythagorean Identities
Start with the side that looks “more complicated”
You may need to rationalize the expression, factor outcommon factors, or multiply all terms by the LCD
Trial and Error (Do whatever it takes)
Once the left side and right side are equal then theequation  is a trigonometric identity
PROVING TRIGONOMETRIC IDENTITIES BY USING BASICIDENTITIES:
Quotient Identity!
Since the left side looks morecomplicated, we will prove theidentity from the left
Cancel out commonfactors
The proof may be the different
but the result is the same
You can also prove the identityfrom the other side
CommonDenominator
Combine the
Fractions
Split into 2separatefractions
PRACTICE: PROVE THE FOLLOWING IDENTITIES:
Factor out sin θ
CommonDenominator
Combine the
Fractions
Get the LCD withthe numerator
PROVING IDENTITIES USING FRACTIONS :
Pythagorean
Identities
Divide thefractions
PROVING IDENTITIES BY FACTORING:
Difference ofsquares
Pythagoreanidentity
CommonDenominator
Expand
Factor out thesine function
PythagoreanIdentity
FOIL
PRACTICE: PROVE THE FOLLOWING IDENTITIES BY FACTORING
Bracket thelast 2 terms
Pythagoreanidentity
Factor: Difference
of squares
PythagoreanIdentity
Factor theTrinomial
PROVE BY CONJUGATING THE EXPRESSION:
Multiply both top &bottom by the conjugate
Pythagoreanidentity
Expand
Quotient Identities
Reciprocal Identity
Pythagorean Identity
Formula Sheet
PRACTICE: PROVE THE FOLLOWING IDENTITY BY CONJUGATINGTHE EXPRESSION:
Multiply both top &bottom by the conjugate
Expand
Pythagoreanidentity
Quotient Identities
Reciprocal Identity
Pythagorean Identity