5.2
Proving
Trigonometric
Identities
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What you’ll learn about
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
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A Proof Strategy
Proving Identities
Disproving Non-Identities
Identities in Calculus
… and why
Proving identities gives you excellent insights into the
way mathematical proofs are constructed.
Copyright © 2011 Pearson, Inc.
Slide 5.2 - 2
General Strategies for Proving an
Identity I
1. The proof begins with the expression on one
side of the identity.
2. The proof ends with the expression on the
other side.
3. The proof in between consists of showing a
sequence of expressions, each one easily
seen to be equivalent to its preceding
expression.
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Slide 5.2 - 3
General Strategies for Proving an
Identity II
1. Begin with the more complicated expression
and work toward the less complicated
expression.
2. If no other move suggests itself, convert the
entire expression to one involving sines and
cosines.
3. Combine fractions by combining them over a
common denominator.
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Slide 5.2 - 4
Example Setting up a Difference of
Squares
Prove the identity:
sin x
1 + cos x
=
.
1 - cos x
sin x
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Slide 5.2 - 5
Example Setting up a Difference of
Squares
Prove the identity:
sin x
1 + cos x
=
.
1 - cos x
sin x
sin x
sin x 1 + cos x
=
×
1 - cos x 1 - cos x 1 + cos x
(sin x )(1 + cos x )
=
1 - cos x
(sin x )(1 + cos x )
=
sin x
1 + cos x
=
sin x
2
2
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Slide 5.2 - 6
General Strategies for Proving an
Identity III
1. Use the algebraic identity (a+b)(a–b) = a2–b2
to set up applications of the Pythagorean
identities.
2. Always be mindful of the “target” expression,
and favor manipulations that bring you closer
to your goal.
Copyright © 2011 Pearson, Inc.
Slide 5.2 - 7
Identities in Calculus
(
)
x = (1 + tan x ) ( sec x )
1. cos x = 1 - sin x ( cos x )
3
2. sec 4
2
2
2
1 1
3. sin x = - cos 2x
2 2
1 1
2
4. cos x = + cos 2x
2 2
5. sin 5 x = 1 - 2 cos 2 x + cos 4 x ( sin x )
2
(
(
)
)
6. sin 2 x cos 5 x = sin 2 x - 2sin 4 x + sin 6 x ( cos x )
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Slide 5.2 - 8
Example Proving an Identity Useful
in Calculus
Prove the following identity:
(
sin 5 x cos 2 x = ( sin x ) cos 2 x - 2 cos 4 x + cos 6 x
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)
Slide 5.2 - 9
Example Proving an Identity Useful
in Calculus
Prove the following identity:
(
sin 5 x cos 2 x = ( sin x ) cos 2 x - 2 cos 4 x + cos 6 x
)
sin 5 x cos 2 x = sin x sin 4 x cos 2 x
( )(
)
= ( sin x ) (1 - cos x ) ( cos x )
= ( sin x ) (1 - 2 cos x + cos x ) ( cos x )
= ( sin x ) ( cos - 2 cos x + cos x )
= ( sin x ) sin x
2
2
cos 2 x
2
2
2
2
2
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4
4
2
6
Slide 5.2 - 10
Quick Review
Write the expression in terms of sines and cosines only.
Express your answer as a single fraction.
1. cot x - tan x
2. sin x sec x - cos x sec x
sin x cos x
3.
+
csc x sec x
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Slide 5.2 - 11
Quick Review
Determine whether or not the equation is an identity.
If not, find a single value of x for which the two
expressions are not equal.
4. ln x = -2 ln x
2
5.
x2 = x
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Slide 5.2 - 12
Quick Review Solutions
Write the expression in terms of sines and cosines only.
Express your answer as a single fraction.
cos 2 x - sin 2 x
1. cot x - tan x
cos x sin x
sin x - cos x
2. sin x sec x - cos x sec x
cos x
sin x cos x
3.
+
1
csc x sec x
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Slide 5.2 - 13
Quick Review Solutions
Determine whether or not the equation is an identity.
If not, find a single value of x for which the two
expressions are not equal.
4. ln x 2 = -2 ln x No, if x were 2 . . .
5.
x2 = x
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No, if x were - 2 . . . , and for any x < 0
Slide 5.2 - 14