Verifying Trigonometric

Identities

What is an Identity?

An identity is a statement that two expressions are equal for every value of the variable.Examples:

xxx 2

The left-hand expression always equals the right-hand expression, no matter what x equals.

The fundamental IdentitiesReciprocal Identities Quotient Identities

xx

xSecx

xx

tan

1cot

cos

1sin

1csc

x

xx

x

xx

sin

coscot

cos

sintan

The beauty of the identities is that we can get all functions in terms of sine and

cosine.

The Fundamental IdentitiesIdentities for Negatives

xx

xx

xx

tan)tan(

)cos()cos(

sin)sin(

The Fundamental Identities

xx

xx

xx

22

22

22

csccot1

sec1tan

1cossin

Pythagorean Identities

The only unique Identity here is the top one, the other two can be obtained using the top

identity.

X

Variations of Identities using Arithmetic

Variations of these Identities

xx

xx22

22

sincos1

cossin1

We can create different versions of many of these identities by using arithmetic.

Let’s look at some examples!

Verifying Trigonometric

Identities

Now we continue on our journey!

An Identity is Not a Conditional Equation

Conditional equations are true only for some values of the variable.

You learned to solve conditional equations in Algebra by “balancing steps,” such as adding the same thing to both sides, or taking the square root of both sides.

We are not “solving” identities so we must approach identities differently.

22

4

82

912

2

2

2

xorx

x

x

x

We Verify (or Prove) Identities by doing the following: Work with one side at a time. We want both sides to be exactly the

same. Start with either side Use algebraic manipulations and/or the

basic trigonometric identities until you have the same expression as on the other side.

Example:xxx cossincot

x

xx

x

xx

cos

sinsin

cos

sincot LHS

and xcos RHS

Since both sides are the same, the identity is verified.

Change everything on both sides tosine and cosine.

Suggestions Start with the more complicated side Try substituting basic identities (changing all

functions to be in terms of sine and cosine may make things easier)

Try algebra: factor, multiply, add, simplify, split up fractions

If you’re really stuck make sure to:

Remember to: Work with only one side at a time!

xx

x

xx

cotsin

1cos

csccos RHS

22 sincoscosecsin Establish the following identity:

In establishing an identity you should NOT move things from one side of the equal sign to the other. Instead substitute using identities you know and simplifying on one side or the other side or both until both sides match.

22 sincoscosecsin Let's sub in here using reciprocal identity

22 sincossin

1sin

22 sincos1

We often use the Pythagorean Identities solved for either sin2 or cos2.

sin2 + cos2 = 1 solved for sin2 is sin2 = 1 - cos2 which is our left-hand side so we can substitute.

22 sinsin

We are done! We've shown the LHS equals the

RHS

cos1

sincotcosec

Establish the following identity:

Let's sub in here using reciprocal identity and quotient identity

Another trick if the denominator is two terms with one term a 1 and the other a sine or cosine, multiply top and bottom of the fraction by the conjugate and then you'll be able to use the Pythagorean Identity on the bottom

We worked on LHS and then RHS but never moved things

across the = sign

cos1

sincotcosec

cos1

sin

sin

cos

sin

1

cos1

sin

sin

cos1

combine fractions

cos1

cos1

cos1

sin

sin

cos1

2cos1

cos1sin

sin

cos1

FOIL denominator

2sin

cos1sin

sin

cos1

sin

cos1

sin

cos1

How to get proficient at verifying identities: Once you have solved an identity go back

to it, redo the verification without looking at how you did it before, this will make you more comfortable with the steps you should take.

Redo the examples done in class using the same approach, this will help you build confidence in your instincts!

Don’t Get Discouraged! Every identity is different Keep trying different approaches The more you practice, the easier it will be

to figure out efficient techniques If a solution eludes you at first, sleep on it!

Try again the next day. Don’t give up! You will succeed!

Establish the identity

Establish the identity

Establish the identity

Homework

14.3 pg 780 #’s 25-28 all, 29-35 odd

AcknowledgementsThis presentation was made possible by

training and equipment from a Merced College Access to Technology grant.

Thank you to Marguerite Smith for the template for some of the slides.