This document discusses proving trigonometric identities. It provides general strategies for proving identities, such as beginning with one side of the identity and manipulating it until reaching the other side. Examples are given of proving specific identities step-by-step and of identities useful in calculus. The document also includes a quick review section testing the manipulation of trigonometric expressions and determining whether equations are identities.

1. 5.2
Proving
Trigonometric
Identities
Copyright © 2011 Pearson, Inc.

2. What you’ll learn about
 A Proof Strategy
 Proving Identities
 Disproving Non-Identities
 Identities in Calculus
… and why
Proving identities gives you excellent insights into the
was mathematical proofs are constructed.
Copyright © 2011 Pearson, Inc. Slide 5.2 - 2

3. General Strategies I for Proving an
Identity
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.
Copyright © 2011 Pearson, Inc. Slide 5.2 - 3

4. General Strategies II for Proving an
Identity
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.
Copyright © 2011 Pearson, Inc. Slide 5.2 - 4

5. Example Setting up a Difference of
Squares
x x
x x
sin 1 cos

Prove the identity: .

1 
cos sin
Copyright © 2011 Pearson, Inc. Slide 5.2 - 5

6. Example Setting up a Difference of
Squares
x x
x x
sin 1 cos

Prove the identity: .

1 
cos sin
x x x
x x x
sin sin 1 
cos
1 cos 1 cos 1 cos
 
  
  
x x
sin 1 cos
2

1 cos
x

  
x x
sin 1 
cos
2
sin
x
x
1 cos
sin
x




Copyright © 2011 Pearson, Inc. Slide 5.2 - 6

7. General Strategies III for Proving an
Identity
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

8. Identities in Calculus
1. cos3 x  1 sin2 xcos x
2. sec4 x  1 tan2  x sec2  x
3. sin2 x 
1
2

1
2
cos2x
4. cos2 x 
1
2

1
2
cos2x
5. sin5 x  1 2cos2 x  cos4  xsin x
6. sin2 x cos5 x  sin2  2sin4 x  sin6  xcos x
Copyright © 2011 Pearson, Inc. Slide 5.2 - 8

9. Example Proving an Identity Useful
in Calculus
Prove the following identity:
sin5 x cos2 x  sin x cos2  2cos4 x  cos6  x
Copyright © 2011 Pearson, Inc. Slide 5.2 - 9

10. Example Proving an Identity Useful
in Calculus
Prove the following identity:
sin5 x cos2 x  sin x cos2  2cos4 x  cos6  x
sin5 x cos2 x  sin x sin4 x cos2 x
 sin x sin2  x2
cos2  x
 sin x 1 cos2  x2
cos2  x
 sin x 1 2cos2 x  cos4  x cos2  x
 sin x cos2  2cos4 x  cos6  x
Copyright © 2011 Pearson, Inc. Slide 5.2 - 10

11. 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 xsec x  cos xsec x
3.
sin x
csc x

cos x
sec x
Copyright © 2011 Pearson, Inc. Slide 5.2 - 11

12. 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  2ln x
5. x2  x
Copyright © 2011 Pearson, Inc. Slide 5.2 - 12

13. Quick Review Solutions
Write the expression in terms of sines and cosines only.
Express your answer as a single fraction.
1. cot x  tan x
cos2 x  sin2 x
cos xsin x
2. sin xsec x  cos xsec x
sin x  cos x
cos x
3.
sin x
csc x

cos x
sec x
1
Copyright © 2011 Pearson, Inc. Slide 5.2 - 13

14. 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  2ln x No
5. x2  x No x  0
Copyright © 2011 Pearson, Inc. Slide 5.2 - 14