math 201 - week 10 · math 201 - week 10 math 201 - week 10 ferenc balogh concordia university 2008...
TRANSCRIPT
![Page 1: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/1.jpg)
MATH 201 - Week 10
MATH 201 - Week 10
Ferenc Balogh
Concordia University
2008 Winter
Based on the textbookJ. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics for Calculus, 5th Edition, Thomson
All figures and videos are made using MAPLE 11 and ImageMagick-convert.
![Page 2: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/2.jpg)
MATH 201 - Week 10
Overview
1 Trigonometric Identities - Section 7.1Trigonometric Identity vs Trigonometric EquationThe Starting Point: Fundamental Trigonometric IdentitiesSimplifying Trigonometric ExpressionsProving Trigonometric Identities
![Page 3: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/3.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Trigonometric Identity vs Trigonometric Equation
Trigonometric Identity
A trigonometric identity is an equation of trigonometric expressionsin θ valid for all values of θ.For example
sin2 θ + cos2 θ = 1
is valid for all θ.
Trigonometric Equation
A trigonometric equation is an equation of trigonometricexpressions in θ valid for some particular values of θ.For example
sin θ + cos θ = 1
is valid for θ = 0 but it is invalid for θ = π4 .
![Page 4: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/4.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Trigonometric Identity vs Trigonometric Equation
Trigonometric Identity
A trigonometric identity is an equation of trigonometric expressionsin θ valid for all values of θ.For example
sin2 θ + cos2 θ = 1
is valid for all θ.
Trigonometric Equation
A trigonometric equation is an equation of trigonometricexpressions in θ valid for some particular values of θ.For example
sin θ + cos θ = 1
is valid for θ = 0 but it is invalid for θ = π4 .
![Page 5: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/5.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
The Starting Point: Fundamental Trigonometric Identities
Fundamental Trigonometric Identities
Reciprocal Identities
csc x =1
sin xsec x =
1
cos xcot x =
1
tan x
tan x =sin x
cos xcot x =
cos x
sin x
Pythagorean Identities
sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x 1 + cot2 x = csc2 x
Even-Odd Identities
sin(−x) = − sin x cos(−x) = cos x tan(−x) = − tan x
![Page 6: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/6.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
The Starting Point: Fundamental Trigonometric Identities
Fundamental Trigonometric Identities
Reciprocal Identities
csc x =1
sin xsec x =
1
cos xcot x =
1
tan x
tan x =sin x
cos xcot x =
cos x
sin x
Pythagorean Identities
sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x 1 + cot2 x = csc2 x
Even-Odd Identities
sin(−x) = − sin x cos(−x) = cos x tan(−x) = − tan x
![Page 7: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/7.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
The Starting Point: Fundamental Trigonometric Identities
Fundamental Trigonometric Identities
Reciprocal Identities
csc x =1
sin xsec x =
1
cos xcot x =
1
tan x
tan x =sin x
cos xcot x =
cos x
sin x
Pythagorean Identities
sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x 1 + cot2 x = csc2 x
Even-Odd Identities
sin(−x) = − sin x cos(−x) = cos x tan(−x) = − tan x
![Page 8: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/8.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
The Starting Point: Fundamental Trigonometric Identities
Cofunction Identities
sin(π
2− x
)= cos x tan
(π2− x
)= cot x sec
(π2− x
)= csc x
cos(π
2− x
)= sin x cot
(π2− x
)= tan x csc
(π2− x
)= sec x
Although these are considered to be fundamental identities, it ismore convenient to prove them later on.
![Page 9: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/9.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Simplifying Trigonometric Expressions
Example. Simplify the trigonometric expression
sin t + cos t cot t.
Solution.
sin t + cos t cot t = sin t + cos tcos t
sin t
=sin2t + cos2 t
sin t
=1
sin t
= csc t.
![Page 10: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/10.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Simplifying Trigonometric Expressions
Example. Simplify the trigonometric expression
sin t + cos t cot t.
Solution.
sin t + cos t cot t = sin t + cos tcos t
sin t
=sin2t + cos2 t
sin t
=1
sin t
= csc t.
![Page 11: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/11.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Simplifying Trigonometric Expressions
Example. Simplify the trigonometric expression
sin t + cos t cot t.
Solution.
sin t + cos t cot t = sin t + cos tcos t
sin t
=sin2t + cos2 t
sin t
=1
sin t
= csc t.
![Page 12: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/12.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Simplifying Trigonometric Expressions
Example. Simplify the trigonometric expression
sin t + cos t cot t.
Solution.
sin t + cos t cot t = sin t + cos tcos t
sin t
=sin2t + cos2 t
sin t
=1
sin t
= csc t.
![Page 13: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/13.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Simplifying Trigonometric Expressions
Example. Simplify the trigonometric expression
sin t + cos t cot t.
Solution.
sin t + cos t cot t = sin t + cos tcos t
sin t
=sin2t + cos2 t
sin t
=1
sin t
= csc t.
![Page 14: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/14.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Simplifying Trigonometric Expressions
Example. Simplify the trigonometric expression
cos x
sin x+
sin x
1 + cos x.
Solution.
cos x
sin x+
sin x
1 + cos x=
cos x(1 + cos x) + sin2 x
sin x(1 + cos x)
=cos x + cos2 x + sin2 x
sin x(1 + cos x)
=cos x + 1
sin x(1 + cos x)
=1
sin x
= csc x .
![Page 15: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/15.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Simplifying Trigonometric Expressions
Example. Simplify the trigonometric expression
cos x
sin x+
sin x
1 + cos x.
Solution.
cos x
sin x+
sin x
1 + cos x=
cos x(1 + cos x) + sin2 x
sin x(1 + cos x)
=cos x + cos2 x + sin2 x
sin x(1 + cos x)
=cos x + 1
sin x(1 + cos x)
=1
sin x
= csc x .
![Page 16: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/16.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Simplifying Trigonometric Expressions
Example. Simplify the trigonometric expression
cos x
sin x+
sin x
1 + cos x.
Solution.
cos x
sin x+
sin x
1 + cos x=
cos x(1 + cos x) + sin2 x
sin x(1 + cos x)
=cos x + cos2 x + sin2 x
sin x(1 + cos x)
=cos x + 1
sin x(1 + cos x)
=1
sin x
= csc x .
![Page 17: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/17.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Simplifying Trigonometric Expressions
Example. Simplify the trigonometric expression
cos x
sin x+
sin x
1 + cos x.
Solution.
cos x
sin x+
sin x
1 + cos x=
cos x(1 + cos x) + sin2 x
sin x(1 + cos x)
=cos x + cos2 x + sin2 x
sin x(1 + cos x)
=cos x + 1
sin x(1 + cos x)
=1
sin x
= csc x .
![Page 18: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/18.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Simplifying Trigonometric Expressions
Example. Simplify the trigonometric expression
cos x
sin x+
sin x
1 + cos x.
Solution.
cos x
sin x+
sin x
1 + cos x=
cos x(1 + cos x) + sin2 x
sin x(1 + cos x)
=cos x + cos2 x + sin2 x
sin x(1 + cos x)
=cos x + 1
sin x(1 + cos x)
=1
sin x
= csc x .
![Page 19: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/19.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Simplifying Trigonometric Expressions
Example. Simplify the trigonometric expression
cos x
sin x+
sin x
1 + cos x.
Solution.
cos x
sin x+
sin x
1 + cos x=
cos x(1 + cos x) + sin2 x
sin x(1 + cos x)
=cos x + cos2 x + sin2 x
sin x(1 + cos x)
=cos x + 1
sin x(1 + cos x)
=1
sin x
= csc x .
![Page 20: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/20.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
How to prove a trigonometric identity?
We transform one side of the equation into the other side by usinga sequence of steps.
Hints
Start with one side (take the more complicated one).
Use the fundamental identities and perform algebraicmanipulations.
In case of an emergency, write all expressions in terms of sinesand cosines only.
PRACTICE!!!
![Page 21: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/21.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
How to prove a trigonometric identity?
We transform one side of the equation into the other side by usinga sequence of steps.
Hints
Start with one side (take the more complicated one).
Use the fundamental identities and perform algebraicmanipulations.
In case of an emergency, write all expressions in terms of sinesand cosines only.
PRACTICE!!!
![Page 22: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/22.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Prove the identity
sec x − cos x
sec x= sin2 x .
rewriting it in terms of sines and cosines.
Solution. The LHS is
LHS =sec x − cos x
sec x
=1
cos x − cos x1
cos x
=1−cos2 x
cos x1
cos x
=1 − cos2 x
cos x· cos x
1
= 1 − cos2 x
= sin2 x
= RHS .
![Page 23: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/23.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Prove the identity
sec x − cos x
sec x= sin2 x .
rewriting it in terms of sines and cosines.Solution. The LHS is
LHS =sec x − cos x
sec x
=1
cos x − cos x1
cos x
=1−cos2 x
cos x1
cos x
=1 − cos2 x
cos x· cos x
1
= 1 − cos2 x
= sin2 x
= RHS .
![Page 24: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/24.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Prove the identity
sec x − cos x
sec x= sin2 x .
rewriting it in terms of sines and cosines.Solution. The LHS is
LHS =sec x − cos x
sec x
=1
cos x − cos x1
cos x
=1−cos2 x
cos x1
cos x
=1 − cos2 x
cos x· cos x
1
= 1 − cos2 x
= sin2 x
= RHS .
![Page 25: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/25.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Prove the identity
sec x − cos x
sec x= sin2 x .
rewriting it in terms of sines and cosines.Solution. The LHS is
LHS =sec x − cos x
sec x
=1
cos x − cos x1
cos x
=1−cos2 x
cos x1
cos x
=1 − cos2 x
cos x· cos x
1
= 1 − cos2 x
= sin2 x
= RHS .
![Page 26: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/26.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Prove the identity
sec x − cos x
sec x= sin2 x .
rewriting it in terms of sines and cosines.Solution. The LHS is
LHS =sec x − cos x
sec x
=1
cos x − cos x1
cos x
=1−cos2 x
cos x1
cos x
=1 − cos2 x
cos x· cos x
1
= 1 − cos2 x
= sin2 x
= RHS .
![Page 27: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/27.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Prove the identity
sec x − cos x
sec x= sin2 x .
rewriting it in terms of sines and cosines.Solution. The LHS is
LHS =sec x − cos x
sec x
=1
cos x − cos x1
cos x
=1−cos2 x
cos x1
cos x
=1 − cos2 x
cos x· cos x
1
= 1 − cos2 x
= sin2 x
= RHS .
![Page 28: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/28.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Prove the identity
sec x − cos x
sec x= sin2 x .
rewriting it in terms of sines and cosines.Solution. The LHS is
LHS =sec x − cos x
sec x
=1
cos x − cos x1
cos x
=1−cos2 x
cos x1
cos x
=1 − cos2 x
cos x· cos x
1
= 1 − cos2 x
= sin2 x
= RHS .
![Page 29: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/29.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
1
sec x + tan x+
1
sec x − tan x= 2 sec x .
Solution.
LHS =1
sec x + tan x+
1
sec x − tan x
=sec x − tan x + sec x + tan x
(sec x + tan x)(sec x − tan x)
=2 sec x
sec2 x − tan2 x
=2 sec x
1= 2 sec x
= RHS .
![Page 30: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/30.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
1
sec x + tan x+
1
sec x − tan x= 2 sec x .
Solution.
LHS =1
sec x + tan x+
1
sec x − tan x
=sec x − tan x + sec x + tan x
(sec x + tan x)(sec x − tan x)
=2 sec x
sec2 x − tan2 x
=2 sec x
1= 2 sec x
= RHS .
![Page 31: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/31.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
1
sec x + tan x+
1
sec x − tan x= 2 sec x .
Solution.
LHS =1
sec x + tan x+
1
sec x − tan x
=sec x − tan x + sec x + tan x
(sec x + tan x)(sec x − tan x)
=2 sec x
sec2 x − tan2 x
=2 sec x
1= 2 sec x
= RHS .
![Page 32: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/32.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
1
sec x + tan x+
1
sec x − tan x= 2 sec x .
Solution.
LHS =1
sec x + tan x+
1
sec x − tan x
=sec x − tan x + sec x + tan x
(sec x + tan x)(sec x − tan x)
=2 sec x
sec2 x − tan2 x
=2 sec x
1= 2 sec x
= RHS .
![Page 33: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/33.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
1
sec x + tan x+
1
sec x − tan x= 2 sec x .
Solution.
LHS =1
sec x + tan x+
1
sec x − tan x
=sec x − tan x + sec x + tan x
(sec x + tan x)(sec x − tan x)
=2 sec x
sec2 x − tan2 x
=2 sec x
1
= 2 sec x
= RHS .
![Page 34: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/34.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
1
sec x + tan x+
1
sec x − tan x= 2 sec x .
Solution.
LHS =1
sec x + tan x+
1
sec x − tan x
=sec x − tan x + sec x + tan x
(sec x + tan x)(sec x − tan x)
=2 sec x
sec2 x − tan2 x
=2 sec x
1= 2 sec x
= RHS .
![Page 35: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/35.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
sin4 t − cos4 t = sin2 t − cos2 t.
Solution.
LHS = sin4 t − cos4 t
= sin4 t − (cos2 t)2
= sin4 t − (1 − sin2 t)2
= sin4 t − (1 − 2 sin2 t + sin4 t)
= sin4 t − 1 + 2 sin2 t − sin4 t
= −1 + 2 sin2 t
= −(sin2 t + cos2 t) + 2 sin2 t
= − sin2 t − cos2 t + 2 sin2 t
= sin2 t − cos2 t
= RHS .
![Page 36: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/36.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
sin4 t − cos4 t = sin2 t − cos2 t.
Solution.
LHS = sin4 t − cos4 t
= sin4 t − (cos2 t)2
= sin4 t − (1 − sin2 t)2
= sin4 t − (1 − 2 sin2 t + sin4 t)
= sin4 t − 1 + 2 sin2 t − sin4 t
= −1 + 2 sin2 t
= −(sin2 t + cos2 t) + 2 sin2 t
= − sin2 t − cos2 t + 2 sin2 t
= sin2 t − cos2 t
= RHS .
![Page 37: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/37.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
sin4 t − cos4 t = sin2 t − cos2 t.
Solution.
LHS = sin4 t − cos4 t
= sin4 t − (cos2 t)2
= sin4 t − (1 − sin2 t)2
= sin4 t − (1 − 2 sin2 t + sin4 t)
= sin4 t − 1 + 2 sin2 t − sin4 t
= −1 + 2 sin2 t
= −(sin2 t + cos2 t) + 2 sin2 t
= − sin2 t − cos2 t + 2 sin2 t
= sin2 t − cos2 t
= RHS .
![Page 38: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/38.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
sin4 t − cos4 t = sin2 t − cos2 t.
Solution.
LHS = sin4 t − cos4 t
= sin4 t − (cos2 t)2
= sin4 t − (1 − sin2 t)2
= sin4 t − (1 − 2 sin2 t + sin4 t)
= sin4 t − 1 + 2 sin2 t − sin4 t
= −1 + 2 sin2 t
= −(sin2 t + cos2 t) + 2 sin2 t
= − sin2 t − cos2 t + 2 sin2 t
= sin2 t − cos2 t
= RHS .
![Page 39: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/39.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
sin4 t − cos4 t = sin2 t − cos2 t.
Solution.
LHS = sin4 t − cos4 t
= sin4 t − (cos2 t)2
= sin4 t − (1 − sin2 t)2
= sin4 t − (1 − 2 sin2 t + sin4 t)
= sin4 t − 1 + 2 sin2 t − sin4 t
= −1 + 2 sin2 t
= −(sin2 t + cos2 t) + 2 sin2 t
= − sin2 t − cos2 t + 2 sin2 t
= sin2 t − cos2 t
= RHS .
![Page 40: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/40.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
sin4 t − cos4 t = sin2 t − cos2 t.
Solution.
LHS = sin4 t − cos4 t
= sin4 t − (cos2 t)2
= sin4 t − (1 − sin2 t)2
= sin4 t − (1 − 2 sin2 t + sin4 t)
= sin4 t − 1 + 2 sin2 t − sin4 t
= −1 + 2 sin2 t
= −(sin2 t + cos2 t) + 2 sin2 t
= − sin2 t − cos2 t + 2 sin2 t
= sin2 t − cos2 t
= RHS .
![Page 41: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/41.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
sin4 t − cos4 t = sin2 t − cos2 t.
Solution.
LHS = sin4 t − cos4 t
= sin4 t − (cos2 t)2
= sin4 t − (1 − sin2 t)2
= sin4 t − (1 − 2 sin2 t + sin4 t)
= sin4 t − 1 + 2 sin2 t − sin4 t
= −1 + 2 sin2 t
= −(sin2 t + cos2 t) + 2 sin2 t
= − sin2 t − cos2 t + 2 sin2 t
= sin2 t − cos2 t
= RHS .
![Page 42: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/42.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
sin4 t − cos4 t = sin2 t − cos2 t.
Solution.
LHS = sin4 t − cos4 t
= sin4 t − (cos2 t)2
= sin4 t − (1 − sin2 t)2
= sin4 t − (1 − 2 sin2 t + sin4 t)
= sin4 t − 1 + 2 sin2 t − sin4 t
= −1 + 2 sin2 t
= −(sin2 t + cos2 t) + 2 sin2 t
= − sin2 t − cos2 t + 2 sin2 t
= sin2 t − cos2 t
= RHS .
![Page 43: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/43.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
sin4 t − cos4 t = sin2 t − cos2 t.
Solution.
LHS = sin4 t − cos4 t
= sin4 t − (cos2 t)2
= sin4 t − (1 − sin2 t)2
= sin4 t − (1 − 2 sin2 t + sin4 t)
= sin4 t − 1 + 2 sin2 t − sin4 t
= −1 + 2 sin2 t
= −(sin2 t + cos2 t) + 2 sin2 t
= − sin2 t − cos2 t + 2 sin2 t
= sin2 t − cos2 t
= RHS .
![Page 44: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/44.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
sin4 t − cos4 t = sin2 t − cos2 t.
Solution.
LHS = sin4 t − cos4 t
= sin4 t − (cos2 t)2
= sin4 t − (1 − sin2 t)2
= sin4 t − (1 − 2 sin2 t + sin4 t)
= sin4 t − 1 + 2 sin2 t − sin4 t
= −1 + 2 sin2 t
= −(sin2 t + cos2 t) + 2 sin2 t
= − sin2 t − cos2 t + 2 sin2 t
= sin2 t − cos2 t
= RHS .
![Page 45: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/45.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
1 + tan2 u
1 − tan2 u=
1
cos2 u − sin2 u.
Solution.
LHS =1 + tan2 u
1 − tan2 u=
1 +(
sin ucos u
)2
1 −(
sin ucos u
)2
=cos2 u+sin2 u
cos2 ucos2 u−sin2 u
cos2 u
=cos2 u + sin2 u
cos2 u· cos2 u
cos2 u − sin2 u
=cos2 u + sin2 u
cos2 u − sin2 u
=1
cos2 u − sin2 u= RHS .
![Page 46: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/46.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
1 + tan2 u
1 − tan2 u=
1
cos2 u − sin2 u.
Solution.
LHS =1 + tan2 u
1 − tan2 u=
1 +(
sin ucos u
)2
1 −(
sin ucos u
)2
=cos2 u+sin2 u
cos2 ucos2 u−sin2 u
cos2 u
=cos2 u + sin2 u
cos2 u· cos2 u
cos2 u − sin2 u
=cos2 u + sin2 u
cos2 u − sin2 u
=1
cos2 u − sin2 u= RHS .
![Page 47: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/47.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
1 + tan2 u
1 − tan2 u=
1
cos2 u − sin2 u.
Solution.
LHS =1 + tan2 u
1 − tan2 u=
1 +(
sin ucos u
)2
1 −(
sin ucos u
)2
=cos2 u+sin2 u
cos2 ucos2 u−sin2 u
cos2 u
=cos2 u + sin2 u
cos2 u· cos2 u
cos2 u − sin2 u
=cos2 u + sin2 u
cos2 u − sin2 u
=1
cos2 u − sin2 u= RHS .
![Page 48: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/48.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
1 + tan2 u
1 − tan2 u=
1
cos2 u − sin2 u.
Solution.
LHS =1 + tan2 u
1 − tan2 u=
1 +(
sin ucos u
)2
1 −(
sin ucos u
)2
=cos2 u+sin2 u
cos2 ucos2 u−sin2 u
cos2 u
=cos2 u + sin2 u
cos2 u· cos2 u
cos2 u − sin2 u
=cos2 u + sin2 u
cos2 u − sin2 u
=1
cos2 u − sin2 u= RHS .
![Page 49: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/49.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
1 + tan2 u
1 − tan2 u=
1
cos2 u − sin2 u.
Solution.
LHS =1 + tan2 u
1 − tan2 u=
1 +(
sin ucos u
)2
1 −(
sin ucos u
)2
=cos2 u+sin2 u
cos2 ucos2 u−sin2 u
cos2 u
=cos2 u + sin2 u
cos2 u· cos2 u
cos2 u − sin2 u
=cos2 u + sin2 u
cos2 u − sin2 u
=1
cos2 u − sin2 u= RHS .
![Page 50: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/50.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
1 + tan2 u
1 − tan2 u=
1
cos2 u − sin2 u.
Solution.
LHS =1 + tan2 u
1 − tan2 u=
1 +(
sin ucos u
)2
1 −(
sin ucos u
)2
=cos2 u+sin2 u
cos2 ucos2 u−sin2 u
cos2 u
=cos2 u + sin2 u
cos2 u· cos2 u
cos2 u − sin2 u
=cos2 u + sin2 u
cos2 u − sin2 u
=1
cos2 u − sin2 u= RHS .
![Page 51: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/51.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
tan v sin v
tan v + sin v=
tan v − sin v
tan v sin v
Solution.
LHS =tan v sin v
tan v + sin v=
sin vcos v sin v
sin vcos v + sin v
=sin2 vcos v
sin v+sin v cos vcos v
=sin2 v
cos v· cos v
sin v + sin v cos v
=sin2 v
sin v + sin v cos v
=sin v
1 + cos v
![Page 52: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/52.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
tan v sin v
tan v + sin v=
tan v − sin v
tan v sin v
Solution.
LHS =tan v sin v
tan v + sin v=
sin vcos v sin v
sin vcos v + sin v
=sin2 vcos v
sin v+sin v cos vcos v
=sin2 v
cos v· cos v
sin v + sin v cos v
=sin2 v
sin v + sin v cos v
=sin v
1 + cos v
![Page 53: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/53.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
tan v sin v
tan v + sin v=
tan v − sin v
tan v sin v
Solution.
LHS =tan v sin v
tan v + sin v=
sin vcos v sin v
sin vcos v + sin v
=sin2 vcos v
sin v+sin v cos vcos v
=sin2 v
cos v· cos v
sin v + sin v cos v
=sin2 v
sin v + sin v cos v
=sin v
1 + cos v
![Page 54: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/54.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
tan v sin v
tan v + sin v=
tan v − sin v
tan v sin v
Solution.
LHS =tan v sin v
tan v + sin v=
sin vcos v sin v
sin vcos v + sin v
=sin2 vcos v
sin v+sin v cos vcos v
=sin2 v
cos v· cos v
sin v + sin v cos v
=sin2 v
sin v + sin v cos v
=sin v
1 + cos v
![Page 55: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/55.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
tan v sin v
tan v + sin v=
tan v − sin v
tan v sin v
Solution.
LHS =tan v sin v
tan v + sin v=
sin vcos v sin v
sin vcos v + sin v
=sin2 vcos v
sin v+sin v cos vcos v
=sin2 v
cos v· cos v
sin v + sin v cos v
=sin2 v
sin v + sin v cos v
=sin v
1 + cos v
![Page 56: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/56.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Example. Verify the identity
tan v sin v
tan v + sin v=
tan v − sin v
tan v sin v
Solution.
LHS =tan v sin v
tan v + sin v=
sin vcos v sin v
sin vcos v + sin v
=sin2 vcos v
sin v+sin v cos vcos v
=sin2 v
cos v· cos v
sin v + sin v cos v
=sin2 v
sin v + sin v cos v
=sin v
1 + cos v
![Page 57: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/57.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Solution (cont).
RHS =tan v − sin v
tan v sin v=
sin vcos v − sin v
sin vcos v sin v
=sin v−sin v cos v
cos vsin v cos v
cos v
=sin v − sin v cos v
cos v· cos v
sin2 v
=1 − cos v
sin v
=1 − cos v
sin v· 1 + cos v
1 + cos v
=1 − cos2 v
sin v(1 + cos v)
=sin2 v
sin v(1 + cos v)=
sin v
1 + cos v.
![Page 58: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/58.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Solution (cont).
RHS =tan v − sin v
tan v sin v=
sin vcos v − sin v
sin vcos v sin v
=sin v−sin v cos v
cos vsin v cos v
cos v
=sin v − sin v cos v
cos v· cos v
sin2 v
=1 − cos v
sin v
=1 − cos v
sin v· 1 + cos v
1 + cos v
=1 − cos2 v
sin v(1 + cos v)
=sin2 v
sin v(1 + cos v)=
sin v
1 + cos v.
![Page 59: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/59.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Solution (cont).
RHS =tan v − sin v
tan v sin v=
sin vcos v − sin v
sin vcos v sin v
=sin v−sin v cos v
cos vsin v cos v
cos v
=sin v − sin v cos v
cos v· cos v
sin2 v
=1 − cos v
sin v
=1 − cos v
sin v· 1 + cos v
1 + cos v
=1 − cos2 v
sin v(1 + cos v)
=sin2 v
sin v(1 + cos v)=
sin v
1 + cos v.
![Page 60: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/60.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Solution (cont).
RHS =tan v − sin v
tan v sin v=
sin vcos v − sin v
sin vcos v sin v
=sin v−sin v cos v
cos vsin v cos v
cos v
=sin v − sin v cos v
cos v· cos v
sin2 v
=1 − cos v
sin v
=1 − cos v
sin v· 1 + cos v
1 + cos v
=1 − cos2 v
sin v(1 + cos v)
=sin2 v
sin v(1 + cos v)=
sin v
1 + cos v.
![Page 61: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/61.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Solution (cont).
RHS =tan v − sin v
tan v sin v=
sin vcos v − sin v
sin vcos v sin v
=sin v−sin v cos v
cos vsin v cos v
cos v
=sin v − sin v cos v
cos v· cos v
sin2 v
=1 − cos v
sin v
=1 − cos v
sin v· 1 + cos v
1 + cos v
=1 − cos2 v
sin v(1 + cos v)
=sin2 v
sin v(1 + cos v)=
sin v
1 + cos v.
![Page 62: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/62.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Solution (cont).
RHS =tan v − sin v
tan v sin v=
sin vcos v − sin v
sin vcos v sin v
=sin v−sin v cos v
cos vsin v cos v
cos v
=sin v − sin v cos v
cos v· cos v
sin2 v
=1 − cos v
sin v
=1 − cos v
sin v· 1 + cos v
1 + cos v
=1 − cos2 v
sin v(1 + cos v)
=sin2 v
sin v(1 + cos v)=
sin v
1 + cos v.
![Page 63: MATH 201 - Week 10 · MATH 201 - Week 10 MATH 201 - Week 10 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5e8bdc13c74c38b9760c00/html5/thumbnails/63.jpg)
MATH 201 - Week 10
Trigonometric Identities - Section 7.1
Proving Trigonometric Identities
Solution (cont).
RHS =tan v − sin v
tan v sin v=
sin vcos v − sin v
sin vcos v sin v
=sin v−sin v cos v
cos vsin v cos v
cos v
=sin v − sin v cos v
cos v· cos v
sin2 v
=1 − cos v
sin v
=1 − cos v
sin v· 1 + cos v
1 + cos v
=1 − cos2 v
sin v(1 + cos v)
=sin2 v
sin v(1 + cos v)=
sin v
1 + cos v.