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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.
MATH 201 - Week 10
Overview
1 Trigonometric Identities - Section 7.1Trigonometric Identity vs Trigonometric EquationThe Starting Point: Fundamental Trigonometric IdentitiesSimplifying Trigonometric ExpressionsProving Trigonometric Identities
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 .
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 .
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
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
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
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.
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.
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.
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.
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.
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.
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 .
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 .
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 .
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 .
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 .
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 .
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!!!
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!!!
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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
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
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
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
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
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
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.
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.
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.
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.
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.
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.
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.
