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Advanced Functions and Modeling
Homework - Unit 6.4
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Simplify the trigonometric expression by following the indicated direction.
1) Rewrite in terms of sine and cosine: tan x · cot x
Establish the identity. (This assignment must be shown to the teacher for full credit.)
cot2 x
1 + sin x
2)
=
csc x - 1
sin x
1)
2)
3) ln cot u = ln cos u - ln sin u
3)
4) csc 4 x - cot 4 x = csc 2 x + cot 2 x
4)
5) (sec u - tan u)(sec u + tan u) = 1
5)
sin x
sin x
+
= 2 tan 2 x
csc x - 1 csc x + 1
6)
6)
7) 1 + sec2 x sin2 x = sec2 x
7)
8) (cos α + sin β)2 + (cos α - sin β)2 = 2(cos2 α + sin2 β)
8)
9)
sin x
sin x
+
= 2 csc x
1 - cos x 1 + cos x
9)
1 + sin v
10) (sec v + tan v)2 =
1 - sin v
10)
1
Answer Key
Testname: UNTITLED12
1) 1
2)
cot2 x
csc2 x -1 (csc x - 1)(csc x +1)
1 + sin x
.
=
=
= csc x +1 =
csc x - 1
csc x - 1
csc x -1
sin x
3) ln cot u = ln
cos u
= ln cos u - ln sin u
sin u
4) csc 4 x - cot 4 x = (csc 2 x + cot 2 x)(csc 2 x - cot 2 x) = (csc 2 x + cot 2 x)(1) = csc 2 x + cot 2 x.
5) (sec u - tan u)(sec u + tan u) = sec2 u - tan2 u = 1
6)
sin x
sin x
(csc x + 1) sin x + (csc x - 1) sin x 1 + sin x + 1 - sin x
+
=
=
= 2 tan2 x.
csc x - 1 csc x + 1
csc2 x - 1
cot2 t
sin2 x
7) 1 + sec2 x sin2 x = 1 +
= 1 + tan2 x = sec2 x.
cos2 x
8) (cos α + sin β)2 + (cos α - sin β)2 = cos2 α + 2 cos α sin β + sin2 β + cos2 α - 2 cos α sin β + sin2 β = 2(cos2 α + sin2 β)
sin x
sin x
sin x[(1 + cos x)+(1 - cos x)]
2 sin x
2 sin x
9)
+
=
=
=
= 2 csc x.
1 - cos x 1 + cos x
(1 - cos x)(1 + cos x)
1 - cos2 x
sin2 x
10) (sec v + tan v)2 = sec2 v + 2 sec v tan v + tan2 v =
1
2 sin v sin2 v
1 + 2 sin v + sin2 v (1 + sin v)2
+
+
=
=
=
cos2 v cos2 v
cos2 v
cos2 v
1 - sin2 v
(1 + sin v)2
1 + sin v
=
(1 - sin v)(1 + sin v) 1 - sin v
2