Download FLEX Mathematics Trigonometry Proving Identities: Angle Addition

FLEX Mathematics
Trigonometry
Proving Identities: Angle Addition and Product Formulas
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Prove the identity.
sin
1 + cos
+
1)
1 + cos
sin
2)
= 2 csc
1)
sin x - cos x sin 2x
=
sec x - csc x
2
2)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the product-to-sum or sum-to-product identities to find an identity for the given expression.
3) sin 11 - sin 19
A) cos 15 sin 4
B) -2 cos 15 cos 4
C) -2 cos 15 sin 4
D) 2 cos
4) sin 17 - sin 5
11
A) -2 cos
2
15
2
3)
sin 2
4)
sin 3
B) 2 cos 11 sin 6
C) -2 cos 11 cos 6
D) cos 11 sin 6
5) sin 7 + sin 19
13
13
sin
A) 2 cos
2
2
5)
B) 2 sin 13 cos 6
C) -2 cos 13 cos 6
D) cos 13 sin 6
6) cos 7 + cos 15
11
cos 2
A) 2 cos
2
6)
B) 2 cos 11 cos 4
C) -2 sin 11 cos 4
D) cos 11 sin 4
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Prove the identity.
7) csc 2 =
8)
1
1 - (sin
7)
- cos )2
sin (u - v)
= sec u cos v - csc u sin v
sin u cos u
8)
1
9)
cos 3t - cos t
= -4 (cos2 t + cos t)
1 - cos t
10) (cot )( cot
9)
- tan ) = cos 2 csc2
10)
11)
cot2 x
1 + sin x
=
csc x - 1
sin x
11)
12)
1 + sin2 x
= 2sec2 - 1
2
cos x
12)
13)
sin(u - v)
= 1 - tan v cot u
cos v sin u
13)
14)
sec x + csc x
= cos x + sin x
tan x + cot x
14)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the product-to-sum or sum-to-product identities to find an identity for the given expression.
15) 4 sin 12 cos 17
A) 2(sin 5 - cos 5 )
B) 2(sin 29 - sin 5 )
C) 2(cos 29 + sin 5 )
D) -2(sin 29 + sin 5 )
15)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Prove the identity.
16) sec 2 =
csc2
2cot2 - csc2
16)
17)
tan 4t - tan 2t
2 tan t
=
1 + tan 4t tan 2t 1 - tan2 t
17)
18)
1 + cos x
= (csc x + cot x)2
1 - cos x
18)
19)
cos3 + sin3
cos + sin
19)
=
2 - sin 2
2
20) sec4 x - tan4 x - 2 tan2 x = 1
20)
2
Answer Key
Testname: ANGLE ADDTION AND PRODUCT IDENTITIESTST
1) Answers may vary. One possible solution is:
sin
1 + cos
sin2 + (1 + cos )2 sin2 + 1 + 2 cos + cos2
+
=
=
1 + cos
sin
(1 + cos )(sin )
(1 + cos )(sin )
=
2(1 + cos )
= 2 csc .
(1 + cos )(sin )
2) Answers may vary. One possible solution is:
sin x - cos x (sec x - csc x)(sin x cos x)
sin 2x
.
=
= sin x cos x =
sec x - csc x
(sec x - csc x)
2
3)
4)
5)
6)
7)
C
B
B
B
Answers may vary. One possible solution is:
1
1
1
csc 2 =
=
=
sin 2
2 sin cos
2
+ 2 sin cos
1 - sin
- cos2
=
1
1 - (sin
- cos )2
.
8) Answers may vary. One possible solution is:
sin (u - v)
sin u cos v - sin v cos u
=
= sec u cos v - csc u sin v.
sin u cos u
sin u cos u
9) Answers may vary. One possible solution is:
cos 3t - cos t cos (2t + t) - cos t cos 2t cos t - sin 2t sin t - cos t (2 cos2 t - 1)(cos t) - 2 sin2 t cos t - cos t
=
=
=
=
1 - cos t
1 - cos t
1 - cos t
1 - cos t
(2 cos t)(cos2 t - sin2 t - 1)
(2 cos t)(cos2 t - (1 - cos2 t) - 1)
4cos t(cos2 t - 1)
=
=
= -4 cos t (cos t + 1)
1 - cos t
1 - cos t
1 - cos t
= -4 (cos2 t + cos t).
10) Answers may vary. One possible solution is:
cos2
(cot )( cot - tan ) = cot2 - 1 =
-1
sin2
=
cos2
- sin2
sin2
= (cos2
- sin2 ) ·
1
sin2
= cos 2 csc2 .
11) Answers may vary. One possible solution is:
cot2 x
csc2 x - 1
1
1 + sin x
=
= csc x + 1 =
+1=
csc x - 1
csc x - 1
sin x
sin x
12) Answers may vary. One possible solution is:
1 + sin2 x
= sec2 x + tan2 x = sec2 x + (sec2 x - 1) = 2sec2 - 1
2
cos x
13) Answers may vary. One possible solution is:
sin(u - v)
sinu cos v - sin v cos u
sin v cos u
=
=1·
= 1 - tan v cot u
cos v sin u
cos v sin u
cos v sin u
14) Answers may vary. One possible solution is:
sec x + csc x
(1/cos x) + (1/sin x)
=
tan x + cot x (sin x/cos x) + (cos x/sin x)
=
(1/cos x) + (1/sin x)
sin x cos x
sin x + cos x
·
=
(sin x/cos x) + (cos x/sin x) sin x cos x sin2 x + cos2 x
15) B
3
Answer Key
Testname: ANGLE ADDTION AND PRODUCT IDENTITIESTST
16) Answers may vary. One possible solution is:
sec 2 =
1
cos 2
=
1
1
csc2
=
·
2cos2 - 1 2cos2 - 1 csc2
=
csc2
2cot2 - csc2
17) Answers may vary. One possible solution is:
tan 4t - tan 2t
2 tan t
= tan(4t - 2t) = tan 2t =
1 + tan 4t tan 2t
1 - tan2t
18) Answers may vary. One possible solution is:
1 + cos x 1 + cos x 1 + cos x (1 + cos x)2 (1 + cos x)2
1 + cos x 2
=
·
=
=
=
1 - cos x 1 - cos x 1 + cos x
sin x
1 - cos2 x
sin2 x
= (csc x + cot x)2
19) Answers may vary. One possible solution is:
cos3 + sin3
(cos + sin )(cos2 - cos sin
=
cos + sin
cos + sin
= cos2
- cos
sin
+ sin2 = 1 - cos
sin
=
+ sin2 )
2 - 2cos sin
2
20) Answers may vary. One possible solution is:
sec4 x - tan4 x - 2 tan2 x = (sec2 x - tan2 x)(sec2 x + tan2 x) - 2 tan2 x
= 1 · (sec2 x + tan2 x) - 2 tan2 x = sec2 x - tan2 x = 1
4
=
2 - sin 2
2