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Math 1060 Verifying Identities Worksheet
Abigail Friese
Name:______________________________________
Pre-requisite Skill: In order to verify trigonometric identities, it is vital that you have strong algebraic skills in
manipulating rational expressions as well as multiplying and factoring multi-term expressions. If you feel that
you need more practice in these prerequisite algebra skills you can access MyMathLab exercises from many
Pearson books by going to http://www.interactmath.com/ .
These are *free* online practice problems with “Help Me Solve This” and “View an Example” features. No fee
or password is required. However, you cannot save your work from session to session.
After you enter the site, you will need to choose a particular textbook from a pull down menu. For basic
algebraic work, a good starting place is Bittinger: Intermediate Algebra, Concepts and Applications, 8e (this is
the text we currently use for Math 1010 at SLCC). For college algebra concepts you may want to look at
Sullivan: College Algebra, 9e (this is the text we currently use for Math 1050 at SLCC). After you choose your
text, choose the appropriate chapter and section to review the desired concepts.
I Warm-up:
1) Answer the following using at least two complete sentences. What is the difference between an
equation and an identity? "An identity is an equation that is satisfied for all values of the variable
for which both sides are defined." ("Trigonometry", Dugopolski, pg. 93)
An equation is a mathematical statement that shows the equality of two
expressions, but is limited to a finite solution set (or none) for any
variables. (Me.)
2) Write down the definitions of the six trigonometric functions (Hint see section 1.4)
sin(theta)=y/r
cos(theta)=x/r
tan(theta)=y/x
sec(theta)=r/x
csc(theta)=r/y
cot(theta)=x/y
3) Which trigonometric identities follow directly from the definitions of the trig functions?
The reciprocal identities set:
sec(theta)=1/cos(theta)
csc(theta)=1/sin(theta)
cot(theta)=1/tan(theta)
4) Write down a proof of the Fundamental Pythagorean identity.
cos 2 θ + sin 2 θ = 1
cos(theta)*cos(theta)+sin(theta)*sin(theta)=1 (factoring terms)
(x/r)^2+(y/r)^2=1 (identity of cos(theta)=x/r and sin(theta)=y/r
x^2+y^2=r^2 (multiply both sides by r^2)
let A=x, B=y, and C=r, and substitute yields:
A^2+B^2=C^2, which is the Pythagorean theorem. :-)
II Communicating your ideas:
At each step in the verification of the identities below, fill in the blank with an appropriate
justification. An example is provided.
Example:
Prove that
sin 3 ( x) + sin( x) cos 2 ( x)
= tan( x)
cos( x)
Choose the left hand side to “simplify” with algebraic steps and basic trig identities. The final
simplified form should match the right hand side.
Pick *one* side to “simplify”
3
2
sin ( x) + sin( x) cos ( x)
cos( x)
5) Prove that
cot( x) + sin( x) =
Equivalent Statement
2
2
sin( x)(sin ( x) + cos ( x))
cos( x)
sin( x)
=
cos( x)
= tan( x)
=
Rationale
Factor sin(x) out of the
numerator
By
the
fundamental
Pythagorean identity
Basic trig identity for
tangent
1 + cos( x) − cos 2 ( x)
sin( x)
Choose the left hand side to “simplify” with algebraic steps and basic trig identities. The final
simplified form should match the right hand side.
Pick *one* side to “simplify”
cot( x ) + sin( x )
Equivalent Statement
cos( x)
+ sin( x)
sin( x)
cos( x)
sin( x)
=
+ sin( x)
sin( x)
sin( x)
=
Rationale (fill in the blanks)
Expanded using definition of Cot
function.
Multiply sin(x) by sin(x)/sin(x)
(which is 1).
=
cos( x) + sin 2 ( x)
sin( x)
=
cos( x) + (1 − cos 2 ( x))
sin( x)
replace sin^2(x) with 1-cos^2(x)
(identity based on sin^2(x)+cos^2(x)=1)
=
1 + cos( x) − cos 2 ( x)
sin( x)
Terms rearranged into the order
from original equation (right side).
Add fractions with equivalent
denominators.
Now show the entire process by completing the entire table. (You may need to add rows to the table or
perhaps you will complete the proof using fewer rows.)
Prove that
sec x + csc x
= sin x + cos x
tan x + cot x
Pick *one* side to “simplify”
Equivalent Statement
sec x + csc x
tan x + cot x
Rationale
tan x (sec x + csc x)
(tan^2 x + 1)
multiply by tan x
tan x
tan x (sec x + csc x)
(sec^2 x)
Pythagorean identity
tan x (1 + cot x)
(sec x)
cancelling out like terms (sec x) from
numerator and denominator
tan x + 1
sec x
multiply tan x through numerator
cos x * tan x + cos x
given sec x = 1/cos x (reciprocal identity), we
reciprocate and multiply through by cos x
sin x + cos x
given tan x = sin x / cos x, the tan x *
cos x simplifies to sin x
6) It is possible to prove an identity by working separately with the two sides until they can be
shown to be equivalent to the same expression. Please fill in the missing steps in the following
tables.
Equivalent
Statement
2
(1 + cot α ) 2 − 2cot α = cot α + 2 cot α
+ 1 − 2 cot α
2
= cot α + 1
= csc 2 α
Since
Rationale
multiply the terms
of the expression
(1+cot^2(a)).
combine
(eliminate) like
terms.
Pythagorean
Identity
(1 + cot α ) 2 − 2 cot α = csc 2 α and
Equivalent
Statement
1
(1 − cos α )(1 + cos α )
=
1
1 − cos 2 α
=
1
sin 2 α
= csc 2 α
Rationale
multiply the terms
in the denominator
Pythagorean
identity
reciprocal
identity
1
= csc2 α , you have proven
(1 − cos α )(1 + cos α )
(1 + cot α )2 − 2 cot α =
1
(1 − cos α )(1 + cos α )
III Verify Identities:
Please verify the following identities. Be sure to follow the processes illustrated in part II including
providing the appropriate rationale. Don’t skip steps!
7) Prove that
sin 3β
= 4 cos β − sec β
sin β cos β
sin (3B)
sin (B) * cos (B)
sin (3B) = sin (2B + B) =
sin (2B) * cos (B) + sin (B) * cos (2B)
Distributive Property, followed by Sin of a Sum
Identity (all used on numerator of expression)
4 cos^2 (B) * sin (B) - sin (B)
sin (B) * cos (B)
Double Angle Identity (used on numerator,
with denominator added back in)
4 cos^2 (B) - 1
cos (B)
Cancelling out like terms
4 cos^2 (B)
cos (B)
-
1
cos (B)
4 cos (B) - sec (B)
8) Prove that
tan 2 x =
1 - cos (2x)
1 + cos (2x)
Distributive Property
cancelling out like terms, and
Reciprocal Identity
1 − cos 2 x
1 + cos 2 x
1 - cos^2 (x) + sin^2 (x)
1 + cos^2 (x) - sin^2 (x)
Double Angle Identity
cos^2 (x) + sin^2 (x) - cos^2 (x) + sin^2 (x)
cos^2 (x) + sin^2 (x) + cos^2 (x) - sin^2 (x)
Pythagorean Identity
2 * sin^2 (x)
2 * cos^2 (x)
Combine like terms
tan^2 (x)
Simplify fraction, and Basic Identity
This assignment needs to be scanned in and emailed to your instructor by the date indicated on the
class calendar.