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Chapter 4
1. 2
)] + 1
+θ 6
n(θ log 21 θ
[si
= 4 sin
Chapter Four 2 θ)
1
cos
2 θ−
− tan
2θ
(sec
1
SOLUTIONθ)
2
ot
+c
1
(
θ )]
2
− sin
log 1−(1
[ By «Craig»
2. 1
2
1 (sec θ −tan θ −cos θ ) [sin(θ + θ)]
2 2 2
log[1−(1−sin 2 θ )] = +1
2 log 2 16
(1+ cot θ) 4 sin θ
At first glance, this problem can look a little bit complicated
and intimidating. A good way to make it easier to solve is to
break it down:
Step 1- Solve the logarithm.
Step 2- Prove the resulting identity.
4. 1
1
1 (sec 2 θ −tan 2 θ −cos2 θ )
2 2 2
(sec θ − tan θ − cos θ)
log[1−(1−sin 2 θθ)] 2
2
[1− (1− sin )]
(1+ cot θ)
(1+
€
€
€
Step 1- Solve the logartithm.
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There are 3 parts to this logarithm:
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Part A. Base
Part B. Base of Argument
Part C. Exponent of Argument
5. 2
[1−(1−sin θ )]
Step 1- Solve the logartithm.
Part A. Simplify the Base.
6. 2
[1−(1−sin θ )]
Method 2
Method 1
Step 1- Solve the logartithm. Part A. Simplify the Base
There are two methods that can be used to get the simplified
version of this expression:
Method 1
Method 2
7. 2
[1−(1−sin θ )]
Method 1
2
1− (1− sin θ)
Step 1- Solve the logartithm. Part A. Simplify the Base
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8. 2
[1−(1−sin θ )]
Method 1
2
1−1+ sin θ
Step 1- Solve the logartithm. Part A. Simplify the Base
Method 1: Expand the original expression.
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9. 2
[1−(1−sin θ )]
Method 1
2
sin θ
Step 1- Solve the logartithm. Part A. Simplify the Base
Method 1: Simplify the resulting expression
10. 2
[1−(1−sin θ )]
Method 2
2
1− (1− sin θ)
Step 1- Solve the logartithm. Part A. Simplify the Base
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11. 2
[1−(1−sin θ )]
Method 2
2
1− (cos θ)
Step 1- Solve the logartithm. Part A. Simplify the Base
sin2 θ + cos2 θ =1 so that
Method 2: Rearrange the identity
cos 2 θ = 1− sin2 θ and use it to simplify the expression.
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12. 2
[1−(1−sin θ )]
Method 2
2
sin θ
Step 1- Solve the logartithm. Part A. Simplify the Base
Method 2: Simplify the resulting expression
13. 2
[1−(1−sin θ )]
2
2
[1− (1− sin θ)] = sin θ
Step 1- Solve the logartithm. Part A. Simplify the Base
2
sin θ
Therefore, the Base is equal to
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14. 1
2
(1+ cot θ)
Step 1- Solve the logartithm.
Part B. Simplify the Base of Argument.
15. 1
2
(1+ cot θ)
Method 2
Method 1
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Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
There are two methods that can be used to get the simplified
version of this expression:
Method 1
Method 2
16. 1
2
(1+ cot θ)
Method 1
1
2
(1+ cot θ)
€
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
17. 1
2
(1+ cot θ)
Method 1
1
2 2
sin θ cos θ
+
€ 2 2
sin θ sin θ
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
cos2 θ
2
Method 1: Recognize that cot θ is the same thing as and sin2 θ
expand it as such. Also, rewrite “1” so it has the same LCD.
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€ €
18. 1
2
(1+ cot θ)
Method 1
1
2 2
sin θ + cos θ
€ 2
sin θ
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
Method 1: Add the two fractions together to get one fraction.
€
19. 1
2
(1+ cot θ)
Method 1
1
1
€ 2
sin θ
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
sin2 θ + cos2 θ =1)
Method 1: Refer to the Pythagorean identity (
and use it to simplify the numerator of the
fraction in the denominator.
€
€
20. 1
2
(1+ cot θ)
Method 1
2
sin θ
€
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
Method 1: Multiply “1” by the reciprocal of the fraction in the
2
denominator ( sin θ ).
€
21. 1
2
(1+ cot θ)
Method 2
1
2
(1+ cot θ)
€
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
22. 1
2
(1+ cot θ)
Method 2
1
2
csc θ
€
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
Method 2: Recognize that the Pythagorean Identity
csc2 θ − cot 2 θ = 1 applies to the denominator and
simplify it.
€
23. 1
2
(1+ cot θ)
Method 2
1
1
€ 2
sin θ
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
csc2 θ is eqivalent to the reciprocal
Method 2: Recognize that
2
of sin θ ( sin θ ) and rewrite it as such.
1
2
€
€
€€
24. 1
2
(1+ cot θ)
Method 2
2
sin θ
€
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
Method 2: Multiply “1” by the reciprocal of the fraction in the
2
denominator ( sin θ ).
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25. 1
2
(1+ cot θ)
1 2
sin θ
=
2
€(1+ cot θ)
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
Therefore, the Base of the Argument
2
sin θ .
is equal to
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26. 1
1
1 (sec 2 θ −tan 2 θ −cos2 θ )
2 2 2
(sec θ − tan θ − cos θ)
log[1−(1−sin 2 θθ)] 2
2
[1− (1− sin )]
(1+ cot θ)
(1+
€
€
Step 1- Solve the logartithm.
€
So far the logarithm goes from this...
27. 1
1
12 (sec 2 θθ−tan 2 θθ−cos2 θθ))
2 2 2
(sec − tan − cos
log[1−(1−sinθ2 θ )] sin θ
sin2 2
(1+ cot θ)
€
€
Step 1- Solve the logartithm.
€
...to this.
28. 1
1
12 (sec 2 θθ−tan 2 θθ−cos2 θθ))
2 2 2
(sec − tan − cos
log[1−(1−sinθ2 θ )] sin θ
sin2 2
(1+ cot θ)
€
€
Step 1- Solve the logartithm.
€
We can recognize this about the logarithm: The Base is the
same as the Base of the Argument. If this is the case, no
matter what the Exponent of the Argument is, the entire
logarithm is equal to the Exponent of the Argument.
30. 1
2
1 (sec θ −tan θ −cos θ ) [sin(θ + θ)]
2 2 2
log[1−(1−sin 2 θ )] = +1
2 log 2 16
(1+ cot θ) 4 sin θ
Step 1- Solve the logartithm.
Now, we can take a look at the original identity.
Since we have solved the logarithm the identity no
longer looks like this...
31. 2
1 [sin(θ + θ)]
+1
=
2 2 2 log 2 16
(sec θ − tan θ − cos θ) 4 sin θ
€ Step 1- Solve the logartithm.
...it looks like this
32. 2
1 [sin(θ + θ)]
= +1
2 2 2 log 2 16
(sec θ − tan θ − cos θ) 4sin θ
€
Step 2- Prove the identity.
33. 2
1 [sin(θ + θ)]
+1
=
2 2 2 log 2 16
(sec θ − tan θ − cos θ) 4 sin θ
€ Step 2- Prove the identity.
To prove the identity, solve the two sides separately:
Side 1
Side 2
34. 1
2 2 2
(sec θ − tan θ − cos θ)
Step 2- Prove the identity.
Side 1.
35. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 2
Method 1
€
Step 2- Prove the identity. Side 1
There are two methods that can be used to get the simplified
version of this expression:
Method 1
Method 2
36. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
1
€ 2 2 2
(sec θ − tan θ − cos θ)
Step 2- Prove the identity. Side 1
€
37. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
1
2
1 sin θ
€ 2
[( 2 ) − ( 2 ) − cos θ]
cos θ cos θ
Step 2- Prove the identity. Side 1
Method 1: Simplify the first two terms so that they are
expressed in terms of sine and/or cosine.
€
38. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
1
2
€ 1− sin θ 2
[( ) − cos θ]
2
cos θ
Step 2- Prove the identity. Side 1
Method 1: Subtract the two fractions in the denominator to
get one fraction.
€
39. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
1
2
€ cos θ 2
[( 2 ) − cos θ]
cos θ
Step 2- Prove the identity. Side 1
2 2
Method 1: Due to the Pythagorean identity ( sin θ + cos θ =1),
2
cos θ
the fraction in the denominator is simplified to cos θ 2
€
.
€ €
40. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
1
€ 2
(1− cos θ)
Step 2- Prove the identity. Side 1
Method 1: Simplify the fraction in the denominator to “1”.
€
41. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
1
€
2
sin θ
Step 2- Prove the identity. Side 1
2 2
Method 1: Due to the Pythagorean identity ( sin θ + cos θ =1),
2
the denominator is simplified to sin θ .
€ €
€
42. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
2
csc θ
€
Step 2- Prove the identity. Side 1
csc 2 θ
1
Method 1: can also be rewritten as
sin2 θ
€
€
43. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 2
1
€ 2 2 2
(sec θ − tan θ − cos θ)
Step 2- Prove the identity. Side 1
€
44. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 2
1
€
2
(1− cos θ)
Step 2- Prove the identity. Side 1
Method 2: Recognize that the the Pythagorean Identity
sec 2 θ −tan 2 θ =1 applies to the denominator of the
expression and simplify it as such.
€
45. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 2
1
€
2
sin θ
Step 2- Prove the identity. Side 1
2
θ + cos2 θ =1),
Method 2: Using the Pythagorean identity ( sin
2
simplify the denominator to sin θ .
€
€
46. 1
2 2 2
(sec θ − tan θ − cos θ)
2
csc θ
€
Step 2- Prove the identity. Side 1
csc 2 θ
1
Method 2: can also be rewritten as
sin2 θ
€
€
47. 2
1 [sin(θ + θ)]
+1
=
2 2 2 log 2 16
(sec θ − tan θ − cos θ) 4 sin θ
€ Step 2- Prove the identity.
So far the we have solved Side 1 of the identity. Now, it no
longer looks like this...
48. 2
[sin(θ + θ)]
2
csc θ = +1
log 2 16
4 sin θ
€
Step 2- Prove the identity.
...it looks like this
49. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Step 2- Prove the identity.
Side 2.
50. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
Method 1
€
Step 2- Prove the identity. Side 2
There are two methods that can be used to get the simplified
version of this expression:
Method 1
Method 2
51. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2
[sin(θ + θ)]
+1
€ log 2 16
4 sin θ
Step 2- Prove the identity. Side 2
52. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2
[sin(θ + θ)]
+1
€ 4
4 sin θ
Step 2- Prove the identity. Side 2
Method 1: Simplify the logarithm that is an exponent for
one of the terms in the denominator. This can be
done by converting it into a power (ie. 2 x =16 ).
€
53. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2
(2sinθ cosθ)
+1
€ 4
4 sin θ
Step 2- Prove the identity. Side 2
Method 1: The expression inside the brackets can be
recognized as one of the “Double Angle Identity”
and therefore can be simplified to 2sinθ cosθ .(see last slide)
54. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
(2)(sinθ)(cosθ)(2)(sinθ)(cosθ)
+1
€ (2)(2)(sinθ)(sinθ)(sinθ)(sinθ)
Step 2- Prove the identity. Side 2
Method 1: Now both the numerator and the denominator
can be expanded. (Expand as much as possible).
55. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
(2)(sinθ)(cosθ)(2)(sinθ)(cosθ)
+1
€ (2)(2)(sinθ)(sinθ)(sinθ)(sinθ)
Step 2- Prove the identity. Side 2
Method 1: Now we can reduce many parts of the expression
and simplify the remaining terms.
56. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2
(cosθ)
+1
2
€ (sinθ)
Step 2- Prove the identity. Side 2
Method 1: This is the resulting expression.
€
57. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2 2
cos θ sin θ
+2
2
sin θ sin θ
€
Step 2- Prove the identity. Side 2
Method 1: From there we can rewrite “1” as a fraction with
the same denominator as the resulting fraction.
€
58. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2 2
cos θ + sin θ
2
sin θ
€
Step 2- Prove the identity. Side 2
Method 1: After that, it simply becomes a matter of adding
the two fractions together...
59. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
1
2
€
sin θ
Step 2- Prove the identity. Side 2
Method 1: ...and applying the Pythagorean Identity.
60. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2
csc θ
€
Step 2- Prove the identity. Side 2
csc 2 θ
1
Method 1: Fianlly, can also be rewritten as
sin2 θ
€
€
61. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
2
[sin(θ + θ)]
+1
€ log 2 16
4 sin θ
Step 2- Prove the identity. Side 2
62. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
2
[sin(θ + θ)]
+1
€ 4
4 sin θ
Step 2- Prove the identity. Side 2
Method 2: The second method starts out the same as the
first, solving the logarithm in the exponent of the
denominator.
63. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
2
(2sinθ cosθ)
+1
€ 4
4sin θ
Step 2- Prove the identity. Side 2
Method 2: Also the same as the first method, we recognize
the expression inside the brackets as a “Double
Angle Identity”...
64. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
(2)(sinθ)(cosθ)(2)(sinθ)(cosθ)
+1
€ (2)(2)(sinθ)(sinθ)(sinθ)(sinθ)
Step 2- Prove the identity. Side 2
Method 2: The expression, once again, is then expanded;
like terms are reduced; and remaining terms are
simplified to get...
66. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
2
(cosθ)
+1
2
€ (sinθ)
Step 2- Prove the identity. Side 2
Method 2: However, instead of rewriting “1” as a fraction,
due to the fact that (cosθ) is an identity itself,
2
2
(sinθ)
rewrite the expression as...
€
€
67. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
2
cot θ +1
€
Step 2- Prove the identity. Side 2
68. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
2
csc θ
€
Step 2- Prove the identity. Side 2
2 2
Method 2: Now, apply the Pythagorean Identity csc θ − cot θ =1
csc 2 θ .
and rewrite the expression as
€
€
69. 2
2
[sin(θ + θ)]
csc θ
=
+1
log 2 16
4 sin θ
Step 2- Prove the identity.
Now that we have solved Side 2, we can safely say...
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70. Step 2- Prove the identity.
...SINCE...
1 2
= csc θ
2 2 2
(sec θ − tan θ − cos θ)