Remember an identity is an equation that is true for all defined values of a

variable.

We are going to use the identities that we have already established to "prove" or establish other identities. Let's summarize the basic identities we have.

RECIPROCAL IDENTITIES

sin

1cosec

cos

1sec

tan

1cot

QUOTIENT IDENTITIES

cos

sintan

sin

coscot

22 sec 1tan

22 cosec cot1

PYTHAGOREAN IDENTITIES

1 cossin 22

EVEN-ODD IDENTITIES

cotcotsecseccoseccosec

tantancoscossinsin

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

•Get common denominators

•If you have squared functions look for Pythagorean Identities

•Work on the more complex side first

•If you have a denominator of 1 + trig function try multiplying top & bottom by conjugate and use Pythagorean Identity

•When all else fails write everything in terms of sines and cosines using reciprocal and quotient identities

•Have fun with these---it's like a puzzle, can you use identities and algebra to get them to match!

Hints for Establishing Identities

MathXTC

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au