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
1
cosec 
sin 
1
sec  
cos 
1
cot  
tan 
QUOTIENT IDENTITIES
sin 
tan  
cos 
sin 2 
cot  
cos 
sin 
PYTHAGOREAN IDENTITIES
2
2
2
tan   1  sec 
 cos   1
1  cot 2   cosec 2
EVEN-ODD IDENTITIES
sin      sin 
cos    cos 
tan      tan 
cosec    cosec sec    sec 
cot      cot 
Establish the following identity: sin  cosec
Let's sub in here using reciprocal identity
 cos   sin 
2
2
sin  cosec  cos   sin 
 1 
2
2
sin  
  cos   sin 
 sin  
2
We are done!
We've shown the
LHS equals the
RHS
2
1  cos   sin 
2
2
sin   sin 
2
2
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 lefthand side so we can substitute.
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.
sin

Establish the following identity: cosec  cot  
Let's sub in here using reciprocal identity and quotient identity 1  cos 
sin 
We worked on cosec  cot  
1  cos 
LHS and then
RHS but never
1
cos 
sin 


moved things
sin  1  cos  FOIL denominator
across the = sign sin 
1  cos   sin
sin  1  cos  
combine fractions



sin 
cos  1  cos  
11cos
Another trick if the
1  cos  sin  1  cos  
denominator is two terms

2
with one term a 1 and the
sin 
1  cos 
other a sine or cosine,
multiply top and bottom of
1  cos  sin  1  cos  

the fraction by the conjugate
2
sin

sin

and then you'll be able to
use the Pythagorean Identity
on the bottom
1  cos  1  cos 

sin 
sin 
Hints for Establishing Identities
•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!
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 Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au