1. 9.1: Identities and Proofs
Grade Distribution
2nd
5th A 4 10 B 8 C 2 1 D F 6 5
No Shows
100+
Range
30-97
53-100
Avg
75.14
82.04
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9.1: Identities and Proofs

2. Pre-Calculus AB PreAP/Dual, Revised ©2014 9.1: Identities and Proofs
Section 9.1
Pre-Calculus AB PreAP/Dual, Revised ©2014
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9.1: Identities and Proofs

3. 9.1: Identities and Proofs
Review
How can I prove that 𝟏+𝐬𝐢𝐧𝒙− 𝐬𝐢𝐧 𝟐 𝒙 𝐜𝐨𝐬𝒙 equals 𝐜𝐨𝐬 𝒙+𝐭𝐚𝐧 𝒙?
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9.1: Identities and Proofs

4. 9.1: Identities and Proofs
Pythagorean Identities:
sin2x + cos2x = 1
1 + tan2x = sec2x
1 + cot2x = csc2x
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9.1: Identities and Proofs

5. 9.1: Identities and Proofs
Quotient Identity
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9.1: Identities and Proofs

6. 9.1: Identities and Proofs
Reciprocal Identity
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9.1: Identities and Proofs

7. Negative Angle Identity
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9.1: Identities and Proofs

8. 9.1: Identities and Proofs
Pythagorean Identity 1=
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9.1: Identities and Proofs

9. 9.1: Identities and Proofs
Steps
Work at one side at a time.
Identify any opportunities to factor an expression, such as adding fractions, squaring a binomial, etc…
Apply whatever sine and cosine functions are applicable
Simplify the equation by using all applicable theorems and identities
CANCEL, CANCEL, CANCEL
Make sure that A = B
Remember, there is MORE THAN ONE WAY to solve an equation
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9.1: Identities and Proofs

10. 9.1: Identities and Proofs
Example 1
Verify the identity, sin x cot x = cos x
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9.1: Identities and Proofs

11. 9.1: Identities and Proofs
Example 2
Verify the identity, cot x sec x = csc x
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9.1: Identities and Proofs

12. 9.1: Identities and Proofs
Your Turn
Verify the identity, sec y cos y = 1
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9.1: Identities and Proofs

13. 9.1: Identities and Proofs
Example 3
Verify the identity, cot2x sin2x + sin2x = 1
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9.1: Identities and Proofs

14. 9.1: Identities and Proofs
Example 3 – Another Way
Verify the identity, cot2x sin2x + sin2x = 1
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9.1: Identities and Proofs

15. 9.1: Identities and Proofs
Example 4
Verify the identity, (sec x – 1) (sec x + 1) = tan2 x
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9.1: Identities and Proofs

16. 9.1: Identities and Proofs
Example 5
Verify the identity, –tan2 x = (tan2x + 1) (cos2x – 1)
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9.1: Identities and Proofs

17. 9.1: Identities and Proofs
Your Turn
Verify the identity, tan2x sec2x – tan2x = tan4 x
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9.1: Identities and Proofs

18. 9.1: Identities and Proofs
Example 6
Verify the identity, 𝐬𝐢𝐧 𝟐 𝒙 𝐜𝐨𝐬 𝒙 +𝐜𝐨𝐬𝒙=𝐬𝐞𝐜𝒙
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9.1: Identities and Proofs

19. 9.1: Identities and Proofs
Example 7
Verify the identity, 𝐜𝐬𝐜 𝒙−𝐬𝐢𝐧 𝒙=𝐜𝐨𝐬 𝒙 𝐜𝐨𝐭 𝒙
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9.1: Identities and Proofs

20. 9.1: Identities and Proofs
Example 7 – Another Way
Verify the identity, 𝐜𝐬𝐜 𝒙−𝐬𝐢𝐧 𝒙=𝐜𝐨𝐬 𝒙 𝐜𝐨𝐭 𝒙
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9.1: Identities and Proofs

21. 9.1: Identities and Proofs
Your Turn
Verify the identity, 𝐭𝐚𝐧 𝒙+𝐜𝐨𝐭 𝒙=𝐬𝐞𝐜 𝒙 𝐜𝐬𝐜 𝒙
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9.1: Identities and Proofs

22. 9.1: Identities and Proofs
Example 8
Verify the identity, 𝐜𝐨𝐬 𝒙 𝐜𝐨𝐭 𝒙 =𝐬𝐢𝐧 𝒙
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9.1: Identities and Proofs

23. 9.1: Identities and Proofs
Your Turn
Verify the identity, 𝐭𝐚𝐧 𝒙 𝐬𝐞𝐜 𝒙 =𝐬𝐢𝐧 𝒙
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9.1: Identities and Proofs

24. 9.1: Identities and Proofs
Example 9
Verify the identity, 𝐬𝐢𝐧 𝒙 𝐜𝐬𝐜 𝒙−𝐬𝐢𝐧 𝒙 = 𝐜𝐨𝐬 𝟐 𝒙
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9.1: Identities and Proofs

25. 9.1: Identities and Proofs
Example 9
Verify the identity, 𝐜𝐬𝐜 𝒙 𝐬𝐞𝐜 𝒙−𝐭𝐚𝐧 𝒙 𝐬𝐢𝐧 𝒙 =𝐜𝐨𝐭 𝒙
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9.1: Identities and Proofs

26. 9.1: Identities and Proofs
Example 9
Verify the identity, 𝐜𝐬𝐜 𝒙 𝐬𝐞𝐜 𝒙−𝐭𝐚𝐧 𝒙 𝐬𝐢𝐧 𝒙 =𝐜𝐨𝐭 𝒙
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9.1: Identities and Proofs

27. 9.1: Identities and Proofs
Your Turn
Verify the identity, 𝐜𝐬𝐜 𝒙=𝐬𝐢𝐧 𝒙+𝐜𝐨𝐬 𝒙 𝐜𝐨𝐭 𝒙
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9.1: Identities and Proofs

28. 9.1: Identities and Proofs
Assignment
Worksheet
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9.1: Identities and Proofs