Brought to you by Brought to you by Tutorial Services – The Math CenterTutorial Services – The Math Center

Trigonometric IdentitiesTrigonometric Identities

In this workshop we will:In this workshop we will:

Look at basic Identities.Look at basic Identities. How other identities can be derived from How other identities can be derived from

basic identities.basic identities. Look at Sum and Difference Identities.Look at Sum and Difference Identities. Look at Double-Angle and Half-Angle Look at Double-Angle and Half-Angle

Identities.Identities. Look at Product and Sum Identities.Look at Product and Sum Identities. How to develop a strategy for solving How to develop a strategy for solving

trigonometric identities.trigonometric identities.

Basic IdentitiesBasic IdentitiesEvery trigonometric function is related to the other

because they are all defined in terms of the coordinates on a unit circle.

Identities from the Definitions

cos

sintan

sin

coscot

cos

1sec

sin

1csc

Basic IdentitiesBasic Identities

From the basic Identities we can define the Reciprocal Identities

Reciprocal Identities

csc

1sin

sec

1cos

cot

1tan

sin

1csc

cos

1sec

tan

1cot

Basic IdentitiesBasic Identities

The Pythagorean Identities can be derived from the fundamental identity

Pythagorean Identities

1cossin 22 xx

1cossin 22 xx

xx 22 csc1cot

xx 22 sec1tan

Sum and Difference IdentitiesSum and Difference Identities

These identities are used in solving equations and in simplifying expressions.

Sum or Difference (Sines, Cosines)

sinsincoscos)cos(

sinsincoscos)cos(

sincoscossin)sin(

sincoscossin)sin(

Sum and Difference IdentitiesSum and Difference Identities

These identities are used in solving equations and in simplifying expressions.

Sum or Difference (Tangents)

tantan1

tantan)tan(

tantan1

tantan)tan(

Double-Angle and Half-Angle IdentitiesDouble-Angle and Half-Angle Identities

xxx cossin22sin xxx 22 sin211cos22cos

The double-angle and half-angle identities are special cases of those identities.

Double-Angle Identities

x

xx

2tan1

tan22tan

Double-Angle and Half-Angle IdentitiesDouble-Angle and Half-Angle Identities

The double-angle and half-angle identities are special cases of those identities.

Half-Angle Identities

2

cos1

2sin

xx

2

cos1

2cos

xx

x

x

x

xx

cos1

sin

sin

cos1

2tan

Product and Sum IdentitiesProduct and Sum Identities

These identities are used to solve certain problems, but not used as often.

Product-to-Sum Identities

)]sin()[sin(2

1cossin BABABA

)]cos()[cos(2

1sinsin BABABA

)]sin()[sin(2

1sincos BABABA

)]cos()[cos(2

1coscos BABABA

Product and Sum IdentitiesProduct and Sum Identities

These identities are used to solve certain problems, but not used as often.

Sum-to-Product Identities

2

cos2

sin2sinsinyxyx

yx

2

sin2

cos2sinsinyxyx

yx

2

cos2

cos2coscosyxyx

yx

2

sin2

sin2coscosyxyx

yx

Developing StrategyDeveloping Strategy

Verifying identities takes practice! The goal is to prove that both sides are equal to one

another.

1. You may work with one or both sides of the equation.2. Rewrite the expressions in terms of sines and cosines only.3. Other algebraic methods can be used such as factoring,

finding the LCD or cross-multiplying.

Verify that the following is an identity:

xxxx 2sectansinsec1

Developing StrategyDeveloping Strategy

xxxx 2sectansinsec1

xx

xx

x2sec

cos

sinsin

cos

11

xx

x 22

2

seccos

sin1

Solution:

This is now verified by the Pythagorean Identity

xx 22 sec1tan

Developing StrategyDeveloping Strategy

Tip: Basic cross-multiplication can simplify your verification!

Now try some on your own.Verify that the following is an identity:

cos

sin1

sin1

cos

Developing StrategyDeveloping Strategy

Tip: Stick to sines and cosines and work with both sides of the equation. Don’t forget your rules for basic math!

Verify that the following is an identity:

xx

xx 2cotsin

sincsc

Trigonometric IdentitiesTrigonometric Identities Links Links

•Trigonometric Identities HandoutTrigonometric Identities Handout•Trigonometric Formulas HandoutTrigonometric Formulas Handout•Equation of a Circle HandoutEquation of a Circle Handout•Trigonometric Substitution QuizTrigonometric Substitution Quiz•Trigonometric Identities QuizTrigonometric Identities Quiz