7.1 – Basic Trigonometric Identities and

Equations

5.4.3

Trigonometric Identities

Quotient Identities

tanθ=sinθcosθ

cotθ=cosθsinθ

Reciprocal Identities

sinθ=1

cscθcosθ=

1secθ

tanθ=1

cotθ

Pythagorean Identities

sin2+ cos2 = 1 tan2+ 1 = sec2 cot2+ 1 = csc2

sin2= 1 - cos2

cos2 = 1 - sin2

tan2= sec2- 1 cot2= csc2- 1

Do you remember the Unit Circle?

• What is the equation for the unit circle?x2 + y2 = 1

• What does x = ? What does y = ? (in terms of trig functions)

sin2θ + cos2θ = 1

Pythagorean Identity!

Where did our pythagorean identities come from??

Take the Pythagorean Identity and discover a new one!

Hint: Try dividing everything by cos2θ

sin2θ + cos2θ = 1 .cos2θ cos2θ cos2θ tan2θ + 1 = sec2θ

Quotient Identity

ReciprocalIdentityanother

Pythagorean Identity

Take the Pythagorean Identity and discover a new one!

Hint: Try dividing everything by sin2θ

sin2θ + cos2θ = 1 .sin2θ sin2θ sin2θ 1 + cot2θ = csc2θ

Quotient Identity

ReciprocalIdentitya third

Pythagorean Identity

Using the identities you now know, find the trig value.

1.) If cosθ = 3/4, find secθ 2.) If cosθ = 3/5, find cscθ.

€

secθ =1

cosθ=

13

4=

4

3

€

sin2θ + cos2θ =1

sin2θ +3

5

⎛

⎝ ⎜

⎞

⎠ ⎟2

=1

sin2θ =25

25−

9

25

sin2θ =16

25

sinθ = ±4

5

cscθ =1

sinθ=

1

± 45

= ±5

4

3.) sinθ = -1/3, find tanθ

4.) secθ = -7/5, find sinθ€

tan2θ +1 = sec2θ

tan2θ +1 = (−3)2

tan2θ = 8

tan2θ = 8

€

tanθ = 2 2

Identities can be used to simplify trigonometric expressions.

Simplifying Trigonometric Expressions

cosθ+sinθ tanθ

=cosθ +sinθ

sinθcosθ

=cosθ +

sin2θcosθ

=

cos2θ + sin2θcosθ

=1

cosθ

=secθ

a)

Simplify.

b)cot2θ

1−sin2θ

=

cos2θsin2θcos2θ

1

=1

sin2θ

=csc2θ

5.4.5

=cos2θsin2θ

×1

cos2θ

Simplifing Trigonometric Expressions

c) (1 + tan x)2 - 2 sin x sec x

=1+2tanx+tan2x−2sinxcosx

=1+tan2x+2tanx−2tanx

=sec2x

d)cscx

tanx+cotx

=1

sinxsinxcosx

+cosxsinx

=1

sinxsin2x+cos2x

sinxcosx

=1

sinx×

sinxcosx1

=cosx

=1

sinx1

sinxcosx

=(1+tanx)2 −2sinx1

cosx

Simplify each expression.

€

1sinθ

cossinθ

1

sinθ•

sinθ

cosθ

1

cosθ= secθ

€

=cos x1

sin x

⎛

⎝ ⎜

⎞

⎠ ⎟sin x

cos x

⎛

⎝ ⎜

⎞

⎠ ⎟

=1

€

cos xcos x

sin x

⎛

⎝ ⎜

⎞

⎠ ⎟+ sin x

cos2 x

sin x+

sin2 x

sin x

cos2 x + sin2 x

sin x

1

sin x= csc x

Simplifying trig Identity

Example1: simplify tanxcosx

tanx cosxsin xcos x

tanxcosx = sin x

Example2: simplifysec xcsc x

sec xcsc x1sin x

1cos x 1

cos xsinx

1= x

=sin xcos x

= tan x

Simplifying trig Identity

Simplifying trig Identity

Example2: simplify cos2x - sin2x

cos x

cos2x - sin2x

cos xcos2x - sin2x 1 = sec x

ExampleSimplify:

= cot x (csc2 x - 1)

= cot x (cot2 x)

= cot3 x

Factor out cot x

Use pythagorean identity

Simplify

ExampleSimplify:

Use quotient identity

Simplify fraction with LCD

Simplify numerator

= sin x (sin x) + cos xcos x

= sin2 x + (cos x)cos x

cos xcos x

= sin2 x + cos2x

cos x = 1

cos x

= sec x

Use pythagorean identity

Use reciprocal identity

Your Turn!Combine fraction

Simplify the numeratorUse pythagorean identity

Use Reciprocal Identity

Practice

One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this:

sintan

cos

xx

x

1sec

cosx

x

1csc

sinx

x

tan cscSimplify:

sec

x x

x

sin 1cos sin

1cos

xx x

x

substitute using each identity

simplify

1cos

1cos

x

x

1

Another way to use identities is to write one function in terms of another function. Let’s see an example of this:

2

Write the following expression

in terms of only one trig function:

cos sin 1x x This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute.

2 2sin cos 1x x 2 2cos 1 sinx x

2= 1 sin sin 1x x

2= sin sin 2x x

20

(E) Examples

• Prove tan(x) cos(x) = sin(x)

RSLS

xLS

xx

xLS

xxLS

sin

coscos

sin

costan

21

(E) Examples

• Prove tan2(x) = sin2(x) cos-2(x)

LSRS

xRS

x

xRS

x

xRS

xxRS

xxRS

xxRS

2

2

2

2

2

2

2

2

22

tan

cos

sin

cos

sin

cos

1sin

cos

1sin

cossin

22

(E) Examples

• Prove tan

tan sin cosx

x x x

1 1

LS xx

LSx

x xx

LSx

x

x

x

LSx x x x

x x

LSx x

x x

LSx x

LS RS

tantan

sin

cos sincos

sin

cos

cos

sinsin sin cos cos

cos sin

sin cos

cos sin

cos sin

1

1

1

2 2

23

(E) Examples

• Prove sin

coscos

2

11

x

xx

LSx

x

LSx

x

LSx x

x

LS x

LS RS

sin

cos

cos

cos( cos )( cos )

( cos )

cos

2

2

1

1

11 1

1

1