2. CHAPTER 7: Trigonometric Identities, Inverse Functions, and Equations
7.1 Identities: Pythagorean and Sum and Difference
7.2 Identities: Cofunction, Double-Angle, and Half-Angle
7.3 Proving Trigonometric Identities
7.4 Inverses of the Trigonometric Functions
7.5 Solving Trigonometric Equations
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3. 7.3 Proving Trigonometric Identities
Prove identities using other identities.
Use the product-to-sum identities and the sum-to-product identities to derive other identities.
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4. Proving Identities: Method 1
Start with either the left side or the right side of the equation and obtain the other side. For example, suppose you are trying to prove that the equation P = Q is an identity. You might try to produce a string of statements (R1, R2, … or T1, T2, …) like the following, which start with P and end with Q or start with Q and end with P:
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5. Proving Identities: Method 2
Work with each side separately until you obtain the same expression. For example, suppose you are trying to prove that the equation P = Q is an identity. You might try to produce two strings of statements like the following, each ending with the same statement S.
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6. Hints for Proving Identities
Use method 1 or 2.
Work with the more complex side first.
Carry out any algebraic manipulations, such as adding, subtracting, multiplying, or factoring.
Multiplying by 1 can be helpful when rational expressions are involved.
Converting all expressions to sines and cosines is often helpful.
Try something! Put your pencil to work and get involved. You will be amazed at how often this leads to success.
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7. Copyright © 2009 Pearson Education, Inc.
Example
Prove the identity
Solution:
Use method 1, begin with the right side.
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8. Copyright © 2009 Pearson Education, Inc.
Example
Prove the identity
Solution:
Use method 2, begin with the right side.
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9. Copyright © 2009 Pearson Education, Inc.
Example
Solution continued:
Now, work on the left side.
We have obtained the same expression on each side so the proof is complete.
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10. Copyright © 2009 Pearson Education, Inc.
Example
Prove the identity
Solution:
Use method 2, begin with the left side.
Now, work on the right side.
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11. Product-to-Sum Identities
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12. Copyright © 2009 Pearson Education, Inc.
Example
Find an identity for
Solution:
Use:
Here x = 3 and y = 7q.
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13. Sum-to-Product Identities
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14. Copyright © 2009 Pearson Education, Inc.
Example
Find an identity for
Solution:
Use:
Here x = 5 and y = .
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