2. 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.
3. RECIPROCAL IDENTITIES
θ
θ
sin
1
cosec =
θ
θ
cos
1
sec =
θ
θ
tan
1
cot =
QUOTIENT IDENTITIES
θ
θ
θ
cos
sin
tan =
θ
θ
θ
sin
cos
cot =
θθ 22
sec1tan =+
θθ 22
coseccot1 =+
PYTHAGOREAN IDENTITIES
1cossin 22
=+ θθ
EVEN-ODD IDENTITIES
( ) ( ) ( )
( ) ( ) ( ) θθθθθθ
θθθθθθ
cotcotsecseccoseccosec
tantancoscossinsin
−=−=−−=−
−=−=−−=−
4. θθθθ 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.
θθθθ 22
sincoscosecsin =−
Let's sub in here using reciprocal identity
θθ
θ
θ 22
sincos
sin
1
sin =−
θθ 22
sincos1 =−
We often use the Pythagorean Identities solved for either sin2
θ or cos2
θ.
sin2
θ + cos2
θ = 1 solved for sin2
θ is sin2
θ = 1 - cos2
θ which is our left-
hand side so we can substitute.
θθ 22
sinsin =
We are done!
We've shown the
LHS equals the
RHS
5. θ
θ
θθ
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
θ
θ
θθ
cos1
sin
cotcosec
+
=−
θ
θ
θ
θ
θ cos1
sin
sin
cos
sin
1
+
=−
θ
θ
θ
θ
cos1
sin
sin
cos1
+
=
−
combine fractions
−
−
+
=
−
θ
θ
θ
θ
θ
θ
cos1
cos1
cos1
sin
sin
cos1
( )
θ
θθ
θ
θ
2
cos1
cos1sin
sin
cos1
−
−
=
−
FOIL denominator
( )
θ
θθ
θ
θ
2
sin
cos1sin
sin
cos1 −
=
−
θ
θ
θ
θ
sin
cos1
sin
cos1 −
=
−
6. •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
•If there is underoute somewhere try to rationalize the
numerator and denominator.
Hints for Establishing Identities
