Verifying Trig Verifying Trig IdentitiesIdentities

(5.1)(5.1)JMerrill, 2009JMerrill, 2009

(contributions from DDillon)(contributions from DDillon)

Trig IdentitiesIdentity: an equation that is true

for all values of the variable for which the expressions are defined

Ex: or (x + 2) = x + 2Conditional Equation: only true for

some of the valuesEx: tan x = 0 or x2 + 3x + 2 = 0

sinxtanx cosx

Recallsin csc

cos sec

tan cot

y rr yx rr xy xx y

Recall - IdentitiesReciprocal Identities

1cscsin

1seccos

1cottan

Also true:

1sincsc

1cossec

1tancot

Recall - IdentitiesQuotient Identities

sintancos

coscotsin

Fundamental Trigonometric

IdentitiesNegative Identities (even/odd)

sin sin csc csc

tan tan cot cot

cos cos sec sec

These are the only even functions!

Recall - IdentitiesCofunction Identities

sin cos2

cos sin

2

tan cot2

cot tan

2

sec csc2

csc sec2

Recall - IdentitiesPythagorean Identities

2 2sin cos 1

2 2tan 1 sec

2 21 cot csc

Simplifying Trig Expressions

• Strategies• Change all functions to sine and cosine

(or at least into the same function)• Substitute using Pythagorean Identities• Combine terms into a single fraction with

a common denominator• Split up one term into 2 fractions• Multiply by a trig expression equal to 1• Factor out a common factor

Simplifying # 1cot sinx x

cos sinsin

x xx

cossin

xx

sin x

cos x

Simplifying #22cos sin

sinx xx

2 2cos sinsin sin

x xx x

2 2cos sinsin

x xx

1sin

x

csc x

Simplifying #32

2

1 coscos x

x2

2

sincos

xx2tan x

Simplifying #4cos sin tanx x x

sincos sincos

xx xx

2sincoscos

xxx

2 2cos sincos cos

x xx x

1cos

x

sec x

Simplifying #53 2sin cos cos

2

x x x

3 2sin sin cos x x x

2 2sin sin cos x x x

sin x

Proof Strategies• Never cross over the equal sign (you

cannot assume equality)• Transform the more complicated side of

the identity into the simpler side.• Substitute using Pythagorean identities.• Look for opportunities to factor• Combine terms into a single fraction

with a common denominator, or split up a single term into 2 different fractions

• Multiply by a trig expression equal to 1.• Change all functions to sines and

cosines, if the above ideas don’t work.ALWAYS TRY SOMETHING!!!

Example• Prove• 2 fractions that need to be added:• Shortcut:

sin cos csc1 cos sin

sin sin cos 1 cos1 cos sin

2 2sin cos cos1 cos sin

1 cos

1 cos sin

1 cscsin

2 2 2 cos 1 cot cot Show x x x

2 2cos 1 cotx x

2 2cos cscx x

22

1cossinx

x2

2

cossin

xx

2cot x

1 + cot2x = csc2 x

22

1cscsin

xx

2

tan cot Prove tancsc

x x xx

2

tan cotcsc

x xx

2

sin coscos sin

csc

x xx x

x

2 2

2

sin cossin cos

csc

x xx xx

2

1sin cos

1sin

x x

x21 sin

sin cos 1

xx x

sincosxx

tan x2

sin coscos sin

csc

x xx x

x