1. Splash Screen

2. Five-Minute Check (over Lesson 5-1) Then/Now New Vocabulary
Example 1: Verify a Trigonometric Identity
Example 2: Verify a Trigonometric Identity by Combining Fractions
Example 3: Verify a Trigonometric Identity by Multiplying
Example 4: Verify a Trigonometric Identity by Factoring
Example 5: Verify an Identity by Working Each Side Separately
Concept Summary: Strategies for Verifying Trigonometric Identities
Example 6: Determine Whether an Equation is an Identity
Lesson Menu

3. Find the value of the expression using the given information
Find the value of the expression using the given information. If tan θ = , find cot θ. A. B. C. D.
5–Minute Check 1

4. Find the value of the expression using the given information
Find the value of the expression using the given information. If sin θ = and cos θ = , find tan θ. A. B. C. D.
5–Minute Check 2

5. Find the value of the expression using the given information
Find the value of the expression using the given information. If csc θ = 3 and cos θ < 0, find cos θ and tan θ. A. B. C. D.
5–Minute Check 3

6. Simplify csc x – csc x cos 2 x.
A. sin x
B. cos x
C. csc x(1 + sin x)
D. 1 – cos x
5–Minute Check 4

7. If sin θ = 0.59, find
A. −0.81
B. −0.59
C. 0.59
D. 0.81
5–Minute Check 5

8. You simplified trigonometric expressions. (Lesson 5-1)
Verify trigonometric identities.
Determine whether equations are identities.
Then/Now

9. verify an identity
Vocabulary

10. Verify a Trigonometric Identity
Verify that
The left-hand side of this identity is more complicated, so transform that expression into the one on the right.
Pythagorean Identity
Reciprocal Identity
Simplify. 
Example 1

11. Verify a Trigonometric Identity
Answer:
Example 1

12. Verify that 2 – cos2 x = 1 + sin2 x.
A. 2 – cos2x = –(sin2x + 1) + 2 = 1 + sin2x
B. 2 – cos2x = 2 – (sin2x + 1) = 1 + sin2x
C. 2 – cos2x = 2 – (1 + sin2x) + 2 = 1 + sin2x
D. 2 – cos2x = 2 – (1 – sin2x) = 1 + sin2x
Example 1

13. Start with the right hand side of the identity.
Verify a Trigonometric Identity by Combining Fractions
Verify that
The right-hand side of the identity is more complicated, so start there, rewriting each fraction using the common denominator 1 – cos2 x.
Start with the right hand side of the identity.
Common denominator
Distributive Property
Example 2

14. Divide out the common factor of sin x.
Verify a Trigonometric Identity by Combining Fractions
Simplify.
Divide out the common factor of sin x.
Simplify.
Quotient Identity 
Example 2

15. Answer: Verify a Trigonometric Identity by Combining Fractions
Example 2

16. Verify that A. B. C. D.
Example 2

17. Verify a Trigonometric Identity by Multiplying
Verify that
Because the left-hand side of this identity involves a fraction, it is slightly more complicated than the right side. So, start with the left side.
Multiply the numerator and denominator by the conjugate of sec x – 1, which is sec x + 1.
Multiply.
Example 3

18. Multiply by the reciprocal of the denominator.
Verify a Trigonometric Identity by Multiplying
Pythagorean Identity
Quotient Identity
Multiply by the reciprocal of the denominator.
Divide out the common factor of sin x.
Example 3

19. Distributive Property
Verify a Trigonometric Identity by Multiplying
Distributive Property
Rewrite the fraction as the sum of two fractions; Reciprocal Identity.
Divide out the common factor of cos x.
Example 3

20. Quotient Identity  Verify a Trigonometric Identity by Multiplying
Example 3

21. Verify a Trigonometric Identity by Multiplying
Answer:
Example 3

22. Verify that A. B. C. D.
Example 3

23. Verify that cos x sec 2 x tan x – cos x tan3 x = sin x.
Verify a Trigonometric Identity by Factoring
Verify that cos x sec 2 x tan x – cos x tan3 x = sin x.
cos x sec 2 x tan x – cos x tan3 x = cos x tan x (sec2 x – tan2 x)
Start with the left-hand side of the identity. Factor.
Pythagorean Identity
= cos x tan x (1)
Quotient Identity =
Divide out the common factor of cos x.
= sin x 
Example 4

24. Verify a Trigonometric Identity by Factoring
Answer: cos x sec 2 x tan x – cos x tan3 x = cos x tan x (sec2 x – tan2 x) = cos x tan x (1) = = sin x
Example 4

25. Verify that csc x – cos x csc x – cos x cot x + cot x = sin x.B. C. D.
Example 4

26. Write as the sum of two fractions.
Verify an Identity by Working Each Side Separately
Verify that
Both sides look complicated, but there is a clear first step for the expression on the left. So, start with the expression on the left.
Write as the sum of two fractions.
Simplify and apply a Reciprocal Identity.
Example 5

27. Verify an Identity by Working Each Side Separately
From here, it is unclear how to transform 1 + cot x into , so start with the right side and work to transform it into the intermediate form 1 + cot x.
Pythagorean Identity
Simplify.
Factor.
Example 5

28. Divide out the common factor of 1 – cot x.
Verify an Identity by Working Each Side Separately
Divide out the common factor of 1 – cot x.
To complete the proof, work backward to connect the two parts of the proof.
Write as the sum of two fractions.
Simplify and apply a Reciprocal Identity.
Multiply by
Example 5

29. Simplify. Pythagorean Identity Simplify.  Answer:
Verify an Identity by Working Each Side Separately
Simplify.
Pythagorean Identity 
Simplify.
Answer:
Example 5

30. Verify that tan2 x – sin2 x = sin2 x tan2 x.B. C. D.
Example 5

31. Key Concept 6

32. Determine Whether an Equation is an Identity
A. Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal.
The equation appears to be an identity because the graphs of the related functions over [–2π, 2π] scl: π by [–1, 3] scl: 1 coincide. Verify this algebraically.
Example 6

33. Divide out the common factor of sec x.
Determine Whether an Equation is an Identity
Pythagorean Identity
Divide out the common factor of sec x.
Example 6

34. Reciprocal Identities
Determine Whether an Equation is an Identity
Reciprocal Identities
Simplify.
Quotient Identity
Answer:
Example 6

35. Determine Whether an Equation is an Identity
B. Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal.
Example 6

36. Determine Whether an Equation is an Identity
The graphs of the related functions do not coincide for all values of x for which the both functions are defined. When , Y1 ≈ 1.43 but Y2 ≈ –0.5. The equation is not an identity.
Example 6

37. Determine Whether an Equation is an Identity
Answer: When , Y1 ≈ 1.43 but Y2 = –0.5. The equation is not an identity.
Example 6

38. B. When , Y1 ≈ 0.71 but Y2 ≈ 0.29. The equation is not an identity.
Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find a value for which both sides are defined but not equal.
A. The equation appears to be an identity because the graphs of the related functions over [–2π, 2π] scl: π by [–3, 3] scl: 1 coincide.
B. When , Y1 ≈ 0.71 but Y2 ≈ The equation is not an identity.
Example 6

39. End of the Lesson