7 – 3:
Trigonometric Identities
Fundamental Identities!
Sum and Difference Identities
sin(−θ ) = − sinθ !
cos(A + B) = cos A cos B − sin A sin B
tan 2 θ + 1 = sec 2 θ !
cos(−θ ) = cosθ !
cos(A − B) = cos A cos B + sin A sin B
1+ cot 2 θ = csc 2 θ !
tan(−θ) = −tan θ !
sin(A + B) = sin A cos B + sin B cos A
sin 2 θ + cos 2 θ = 1
!
!
sin(A − B) = sin A cos B − sin B cos A
!
Double Angle Identities!
Half Angle Identities
sin(2A) = 2sin A cos A!
cos
1 + cos A
⎛ A⎞
=
⎝ 2⎠
2
cos(2A) = cos2 A − sin 2 A
cos(2A) = 1−2 sin 2 A!
sin
1− cos A
⎛ A⎞
=
⎝ 2⎠
2
tan
1− cos A
⎛ A⎞
=
⎝ 2⎠
1 + cos A
cos(2A) = 2cos2 A − 1!
tan(2A) =
2tan A
!
1− tan 2 A
tan 2 θ + 1 = sec 2 θ really means tan 2 θ + 1 ⇔ sec 2 θ
When you see the equal sign in the identity tan2 θ + 1 = sec 2 θ you may state that it means that if you
plug any radian value into the variable θ the equation will be true. That is true if you are
considering the identity as an equation. Equations allow the movement of the terms from one side of
he equal sign to the other if you consider the identity as an equation you could also subtract 1 from
both sides to get a new identity tan 2 θ = sec 2 θ − 1. Neither of the these two interpretations are what
we mean when we state a Trigonometric identity. You could say that there are two expressions, one
on the left hand side of the equals sign (LHS) and the other on the right hand side (RHS) and there
expressions are equivalent expressions. In other words you could use either expression in a problem
if you wanted to. We say that you can substitute one expression for the other. If you see the
expression sec 2 θ − 1 you could substitute tan 2 θ in it place. In the case of identities the equal sign
really means the two statements are separate but equivalent. A better way to state the identity would
2
2
be to use a ⇔ in place of the equal sign , tan θ + 1 ⇔ sec θ but textbooks have chosen to use the
equal sign and expect you to understand what the identity really meas.
if In future courses, especially calculus, you will have a complex statements with several
expressions that are terms or factors in the expression. You will simplify that expression by taking
complex terms or factors and replacing them with simpler equivalent expressions that are found in the
list of trig identities.
Math 370 7 - 3 Trig Identity Lecture!
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© 2015 Eitel
Example
Simplify tan x •
tan x =
1
sin x
sin x
is an idenity
cos x
so we can replace tan x with
sin x
to get
cos x
sin x
1
•
cos x sin x
cancel the comon factor sin x to get
1
cos x
1
= sec x is an idenity
cos x
1
so we can replace
with sec x to get
cos x
sec x
so we know that tan x •
1
= sec x
sin x
1
. How did we know to continue
cos x
to get simplifying to get sec x . That's a good question. To make it easier to know when to stop we
will make it a lot easier in this course. We will tell you what the answer will be when you do all the
correct steps. We do this by writing the expression to be simplified and then setting it equal to what
the reduced answer will be. The process of showing the steps you take to get the answer is called
“proving the identity”.
You may ask why didn't we stop when we got the expression
If I know the answer in advance what does “proving an identity ask me to do”
You will be given a complex expression on one side of the equal sign and told what the simplified
expression will be on the other side. The process of showing all the steps you take to get the answer
in a clear and detailed manner is called “proving the identity”.
In calculus it will be a harder task to reduce an expression. You will not be given the answer as
a help o guide you in the simplification process. You will have to know what the “good “ expressions
we use in the course are and reduce your expression to one of them. For this reason we put the
expression to be reduced on the left hand side and the reduced expression on the right hand side.
Math 370 7 - 3 Trig Identity Lecture!
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What are the steps I am allowed to use to simplify the left side so that it is the same
expression as the right side of the equation.
1. Use the identity list. FInd a complex expression and substitute an alternate expression using a
trigonometric Identity found on he identity lest.
2. Convert everything to sine and cosine. This reduces the number of identities you will need to
consider in the next steps to ones with sine or cosine in them.
3. Factor an expression. Look for a common factor. Look for the difference of squares or a
trinomial that can be factored. The factored expression may have common factors that will cancel
out or a factor may be an expression that can be replaced with a simpler expression found by
using the identity list.
4. Distribute or FOIL. This may create a second degree expression that is on the identities list. All
or party of the new expression may be an expression that can be replaced with a simpler
expression found by using the identity list.
4. Use the identity list. FInd a complex expression and substitute a alternate expression using a
trigonometric Identity the identity lest.
5. Multiply the numerator and denominator of a fraction by a “well chosen 1” term to create an
identity. Use the identity to simplify the expression.
6. Add two fractions to get one fraction. If you have an expression with two separate fractions,
Add the separate fractions by getting a common numerator and denominator. Do this by
multiplying each term by a “well chosen 1” term. Add the numerators to create a useful identity or
an expression that factors. Use the identity to simplify the expression or factor and cancel to reduce
the expression.
7. Make one fraction into two fractions. If you have a fraction that has a polynomial in the
numerator and a monomial in the denominator break the fraction into two separate fractions
each with a common denominator and reduce each separate fraction.
8. Break up a polynomial into more terms so that a part of of the terms in the sum form an identity.
Use the identity to simplify the expression.
Math 370 7 - 3 Trig Identity Lecture!
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© 2015 Eitel
Use the identities and steps listed above to prove each Identity.
Example 1
Convert everything to sine and cosine.
You must show all steps and work so that it is clear what was done on each step.
I did it in my head is not an approved step.
cot x
= cos x
csc x
cos x
sin x
1
sin x
convert to sine and cosine
= cos x
invert and multiply
cos x sin x
•
= cos x
sin x
1
cancel
cos x = cos x
Example 2
Break up a polynomial into more terms so that a part of of the terms in the sum form an identity.
Use the identity to simplify the expression.
2sin 2 x + 3cos 2 x = 2 + cos2 x
break up the polynominal into more terms
2sin 2 x + 2cos 2 x + 1cos2 x = 2 + cos 2 x
factor out a 2 in the first two terms
2(sin 2 x + cos 2 x) + 1cos 2 x = 2 + cos2 x
use sin 2 x + cos2 x = 1
2(1) + cos 2 x = 2 + cos2 x
2 + cos2 x = 2 + cos 2 x
Math 370 7 - 3 Trig Identity Lecture!
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© 2015 Eitel
Example 3
Add separate fractions by getting a common numerator and denominator. Do this by
multiplying each term by a “well chosen 1” term. Add the numerators to create a useful identity or an
expression that factors. Use the identity to simplify the expression or factor and cancel to reduce the
expression. The most common way to lose points on the test is to fail to show all steps in a clear
manner.
1 + sin x cot x − cos x
2
+
=
sin x
cos x
sin x
(state cot x in terms of cos x and sin x )
cos x
− cos x
1 + sin x
2
sin
x
+
=
sin x
cos x
sin x
(multiply each fractions by a "well chosen 1" to get a CD )
cos x
− cos x
cos x 1 + sin x sin x
2
sin
x
•
+
•
=
cos x
sin x
sin x
cos x
sin x
distriubte
cos x + cos x sin x cos x − cos x sin x
+
cos x sin x
cos x sin x
=
2
sin x
(combine the numerators)
cos x + cos x sin x + cos x − cos x sin x
2
=
cos x sin x
sin x
2cos x
2
=
cos x sin x sin x
(cancel)
2
2
=
sin x
sin x
Math 370 7 - 3 Trig Identity Lecture!
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© 2015 Eitel
Example 4
Factor expressions to get common factors that cancel out.
sin 2 x − 1
sin x + 1
=
tan x sin x − tan x
tan x
( factor each expression )
(sin x + 1) (sin x − 1) sin x + 1
=
tan x (sin x − 1)
tan x
(cancel)
sin x + 1
sin x + 1
=
tan x
tan x
Math 370 7 - 3 Trig Identity Lecture!
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© 2015 Eitel
Example 5
Multiply the numerator and denominator by a “well chosen 1” term to create an identity.
Use the identity to simplify the expression.
The most common way to lose points on the test
is to fail to show all steps in a clear manner.
cos x
1− sin x
=
1 + sin x
cos x
multiply the left side of the equation by
1− sin x
1− sin x
distrubite and FOIL
1− sin x
cos x
1− sin x
•
=
1− sin x 1 + sin x
cos x
(1− sin x) cos x
2
1− sin x
=
1− sin x
cos x
use 1− sin 2 x = cos2 x
(1− sin x) cos x
2
cos x
=
1− sin x
cos x
cancel
1− sin x
1− sin x
=
cos x
cos x
Math 370 7 - 3 Trig Identity Lecture!
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Example 6
Add separate fractions by getting a common numerator and denominator. Do this by multiplying
each term by a “well chosen 1” term. Add the numerators to create a useful identity or an expression
that factors. Use the identity to simplify the expression or factor and cancel to reduce the expression.
The most common way to lose points on the test is to fail to show all steps in a clear manner.
sin x
1 + cos x
+
= 2csc x
1 + cos x
sin x
(multiply each fraction by a "well chosen 1" to get a CD )
sin x
sin x
1 + cos x 1 + cos x
•
+
•
= 2csc x
sin x 1 + cos x 1 + cos x
sin x
muiltiply or FOIL the numerators
sin 2 x
1 + 2cos x + cos 2 x
+
sin x(1 + cos x)
sin x(1 + cos x)
= 2csc x
(combine the numerators)
sin 2 x + 1 + 2cos x + cos2 x
= 2csc x
sin x(1 + cos x)
use sin 2 + cos 2 x = 1
1 + 1 + 2cos x
= 2csc x
sin x(1 + cos x)
2 + 2cos x
= 2csc x
sin x(1 + cos x)
(factor and cancel)
2(1 + cos x)
= 2csc x
sin x(1 + cos x)
2
= 2csc x
sin x
use
1
= csc x
sin x
2csc x = 2csc x
Math 370 7 - 3 Trig Identity Lecture!
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© 2015 Eitel
Example 7
Add separate fractions by getting a common numerator and denominator. Do this by multiplying
each term by a “well chosen 1” term. Add the numerators to create a useful identity or an expression
that factors. Use the identity to simplify the expression or factor and cancel to reduce the expression.
The most common way to lose points on the test is to fail to show all steps in a clear manner.
tan x + cot x
=1
sec x csc x
(change to sin x and cos x )
sin x cos x
+
cos x sin x
1
1
•
cos x sin x
=1
sin x cos x
+
cos x sin x
1
cos x sin x
= 1 (invert and mutiply)
⎛ sin x cos x ⎞ cos x sin x
+
•
=1
⎝ cos x sin x ⎠
1
multiply each term in the right bracket by a "well chosen 1" to get a CD
⎛ sin x sin x cos x cos x ⎞ cos x sin x
•
+
•
•
=1
⎝ sin x cos x cos x sin x ⎠
1
muiltiply the numerators
⎛ sin 2 x
cos2 x ⎞ cos x sin x
+
=1
⎜ sin x cos x sin x cos x ⎟ •
1
⎝
⎠
(combine the numerators)
⎛ sin 2 x + cos2 x ⎞ cos x sin x
=1
⎜ sin x cos x ⎟ •
1
⎝
⎠
use sin 2 x + cos2 x = 1
1
⎛
⎞ cos x sin x
•
=1
⎝ sin x cos x ⎠
1
(cancel)
1 =1
Math 370 7 - 3 Trig Identity Lecture!
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© 2015 Eitel
Example 8
Use an identity to simplify the expression.
(
)
tan 2 x 1 + cot 2 x
=
1
1− sin 2 x
(rewrite cot 2 x in terms of tan 2 x )
1 ⎞
1
⎛
tan 2 x ⎜1 +
=
⎟
⎝ tan 2 x ⎠ 1− sin 2 x
distribute
tan 2 x + 1 =
1
1− sin 2 x
use tan 2 x + 1 = sec 2 x
sec 2 x =
1
1− sin 2 x
use sec 2 x =
use sec 2 x =
1
cos2 x
1
cos2 x
sec 2 x = sec 2 x
Math 370 7 - 3 Trig Identity Lecture!
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© 2015 Eitel
Example 9
Write as sin (x) and cos (x)
and break a fraction into 2 separate fractions with a CD
tan x − cot x
= sec 2 x − csc 2 x
sin x cos x
(rewrite cot x and tan x in terms of sinx and cos x )
sin x cos x
−
cos x sin x = sec 2 x − csc 2 x
sin x cos x
Multiply by a "well chosen 1")
sin x cos x
−
⎛ sin x cos x ⎞ cos x sin x
= sec 2 x − csc 2 x
⎝ sin x cos x ⎠ sin x cos x
sin 2 x − cos 2 x
2
2
sin x cos x
= sec 2 x − csc 2 x
the right side has 2 seperate terms so we break the fraction on
the left into 2 seperate fractions each of which has the common denominator
sin 2 x
2
2
sin x cos x
−
cos 2 x
2
2
sin x cos x
= sec 2 x − csc 2 x
reduce each fraction on the left side
1
1
−
= sec 2 x − csc 2 x
2
2
cos x
sin x
use
1
2
cos x
= sec 2 x and
1
2
sin x
= csc 2 x
sec 2 x − csc 2 x = sec 2 x − csc 2 x
Math 370 7 - 3 Trig Identity Lecture!
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© 2015 Eitel