This document provides information about trigonometric identities. It defines an identity as an equation that is true for all values of a variable, where the left and right sides are always equal. It then lists several common trigonometric identities, including reciprocal, quotient, Pythagorean, and even-odd identities. Examples are shown of establishing two identities by substituting known identities and simplifying until both sides are equal. Hints are provided for establishing identities, such as getting common denominators, using Pythagorean identities, and writing everything in terms of sines and cosines.

1. www.PinoyBIX.org
Presents:
Trigonometric Identities

2. Remember an identity is an equation
that is true for all defined values of a
variable.
The left-hand expression always equals the right-hand
expression, no matter what x equals.
xxx 2

3. RECIPROCAL IDENTITIES


sin
1
cosec 


cos
1
sec 


tan
1
cot 
QUOTIENT IDENTITIES



cos
sin
tan 



sin
cos
cot 
 22
sec1tan 
 22
coseccot1 
PYTHAGOREAN IDENTITIES
1cossin 22
 
EVEN-ODD IDENTITIES
     
      

cotcotsecseccoseccosec
tantancoscossinsin



4. Note:
 In every identities we can write every
functions in terms of sine and cosine.
 We can create different versions of
many of these identities by using
arithmetic.

5.  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
sincos
sin
1
sin 





 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

6. 


cos1
sin
cotcosec

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
sin
cotcosec






 cos1
sin
sin
cos
sin
1






cos1
sin
sin
cos1



combine fractions 






















cos1
cos1
cos1
sin
sin
cos1
 




2
cos1
cos1sin
sin
cos1




FOIL denominator
 




2
sin
cos1sin
sin
cos1 






sin
cos1
sin
cos1 



7. 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, you can use
identities and algebra to get them to match!
Hints for Establishing Identities

8. Online Notes and Presentations
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credit: Shawna Haider