Copyright © 2011 Pearson, Inc.

5.2Proving

Trigonometric Identities

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What you’ll learn about

A Proof Strategy Proving Identities Disproving Non-Identities Identities in Calculus

… and whyProving identities gives you excellent insights into the way mathematical proofs are constructed.

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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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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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Example Setting up a Difference of Squares

sin 1 cosProve the identity: .

1 cos sin

x x

x x

+=

−

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Example Setting up a Difference of Squares

sin 1 cosProve the identity: .

1 cos sin

x x

x x

+=

−

( )( )

( )( )2

2

sin sin 1 cos

1 cos 1 cos 1 cossin 1 cos

1 cos

sin 1 cos

sin1 cos

sin

x x x

x x xx x

xx x

xx

x

+= ⋅

− − ++

=−

+=

+=

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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.

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Identities in Calculus

1. cos3 x = 1−sin2 x( ) cosx( )

2. sec4 x= 1+ tan2 x( ) sec2 x( )

3. sin2 x=12−12cos2x

4. cos2 x=12+12cos2x

5. sin5 x= 1−2cos2 x+ cos4 x( ) sinx( )

6. sin2 xcos5 x= sin2 x−2sin4 x+sin6 x( ) cosx( )

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Example Proving an Identity Useful in Calculus

Prove the following identity:

sin5 x cos2 x = sinx( ) cos2 x−2cos4 x+ cos6 x( )

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Example Proving an Identity Useful in Calculus

Prove the following identity:

sin5 x cos2 x = sinx( ) cos2 x−2cos4 x+ cos6 x( )

sin5 x cos2 x =sinxsin4 xcos2 x

= sinx( ) sin2 x( )2cos2 x( )

= sinx( ) 1−cos2 x( )2cos2 x( )

= sinx( ) 1−2cos2 x+ cos4 x( ) cos2 x( )

= sinx( ) cos2−2cos4 x+ cos6 x( )

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Quick Review

Write the expression in terms of sines and cosines only.

Express your answer as a single fraction.

1. cot x−tanx

2. sinxsecx−cosxsecx

3. sinxcscx

+cosxsecx

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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 x2 =−2 lnx

5. x2 =x

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Quick Review Solutions

Write the expression in terms of sines and cosines only.

Express your answer as a single fraction.

1. cot x−tanx cos2 x−sin2 xcosxsinx

2. sinxsecx−cosxsecx sinx−cosx

cosx

3. sinxcscx

+cosxsecx

1

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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 x2 =−2 lnx No, if x were 2 . . .

5. x2 =x No, if x were −2 . . . , and for any x< 0