ExamView - A2.A.77.DoubleAngleIdentities4.tst

ExamView - A2.A.77.DoubleAngleIdentities4.tst

Regents Exam Questions A2.A.77: Double Angle Identities 4
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A2.A.77: Double Angle Identities 4: Apply the double-angle and half-angle formulas for
trigonometric functions
6 For all values of  for which the expressions are
defined, prove that the following is an identity:
2 sin 
1  tan 2  
cos  sin 2
1
sin2x the same expression as sin x? Justify
2
your answer.
1 Is
2 For all values of  for which the expressions are
defined, prove that the following is an identity:
2tan 
sin 2 
1  tan 2 
7 Prove the identity: sin2x  tan x(2  2 sin 2 x)
8 For all values of A for which the expressions are
defined, show that the following is an identity:
2 sin 2 A
 cot A  sec A csc A
sin 2A
3 For all values of  for which the expressions are
defined, prove the following is an identity:
1  tan 2 
cos 2 
1  tan 2 
9 For all values of  for which the expressions are
defined, prove the following is an identity:
cos (90   ) sec 

sin 2
2
4 For all value of x for which the expressions are
defined, prove the following is an identity:
2  2cos 2 x
sec x sin2x 
sin x
10 For all value of x for which the expressions are
defined, prove the following is an identity:
2 sinx cos x  tan2x cos 2x
5 For all values of x for which the expressions are
defined, show that the following equation is an
2tanx
identity: sin 2x 
1  tan 2 x
11 For all values of A for which the expressions are
defined, prove the following is an identity:
1  cos A  cos 2A
cot A 
sinA  sin 2A
1
Regents Exam Questions A2.A.77: Double Angle Identities 4
Name: _______________________
www.jmap.org
12 For all values of  for which the expressions are
defined, prove the following is an identity:
sin 2  sin 
 tan 
cos 2  cos   1
18 Prove the following identity:
sin 
sec 

2
cot 
sin   cos 2
19 Prove the following identity:
(sin x  cos x) 2  1
 (sin 2x)(tan x)(csc x)
cos x
13 For all values of x for which the expressions are
defined, prove that the following is an identity:
tanx  cot x  2 csc 2x
14 For all value of x for which the expression is
defined, prove that the following is an identity:
sin 2x
cot x 
1  cos 2x
15 For all values of x for which the expressions are
defined, prove the following is an identity:
cos 2x sin 2x

 csc x
cos x
sin x
cos 2A  sin 2 A
as a single trigonometric
cos A
function for all values of A for which the fraction is
defined.
16 Express
17 For all value of x for which the expressions are
defined, prove the following is an identity:
cos 2x
 sin x  csc x  sin x
sin x
2
ID: A
A2.A.77: Double Angle Identities 4: Apply the double-angle and half-angle formulas for
trigonometric functions
Answer Section
1 ANS:
Not an identity, so the expressions are not equal.
REF: 060222b
2 ANS:
2 tan 
sin 2 
1  tan 2 
2 sin 
cos 
2 sin  cos  
sec 2 
1
cos 
cos  
1
cos 2 
cos   cos 
REF: 088442siii
1
ID: A
3 ANS:
cos 2 
1  tan 2 
1  tan 2 
sin 2 
cos 2 
sec 2 
1
cos 2 
sin 2 
cos 2 
1
cos 2 
1
cos 2 
ˆ
ÊÁ
ÁÁ
sin 2  ˜˜˜˜
2
Á
cos   sin   ÁÁ 1 
˜˜ cos 
2
˜
Á
cos  ¯
Ë
2
2
cos 2   sin 2   cos 2   sin 2 
REF: 018542siii
4 ANS:
sec x sin2x 
2  2cos 2 x
sin x
2(1  cos 2 x)
1
 2sinx cos x 
cos x
sin x
sin x 
sin 2 x
sinx
sin x  sinx
REF: 068541siii
5 ANS:
2tanx
sin 2x 
1  tan 2 x
2sinx
cos x
2 sin x cos x 
sec 2 x
2sinx
cos x
2 sin x cos x 
1
cos 2 x
2sinx 2sinx

cos x
cos x
REF: 068738siii
2
ID: A
6 ANS:
sec 2  
2 sin 
cos   2 sin   cos 
1
1

2
cos  cos 2 
REF: 088737siii
7 ANS:
sin2x  tanx(2  2sin 2 x)
2 sinx cos x 
sinx
(2(1  sin 2 x))
cos x
(2(cos 2 x))
2 cos x 
cos x
cos x  cos x
REF: 018941siii
8 ANS:
2sin 2 A
 cot A  sec A csc A
sin2A
2 sin 2 A
cos A
1
1



2 sin A cos A sinA cos A sin A
sin A cos A
1
1



cos A sinA cos A sin A
sin 2 A  cos 2 A
1

cos A  sinA
cos A  sin A
1
1

cos A  sinA cos A  sin A
REF: 068942siii
9 ANS:
cos (90   ) sec 

2
sin 2
cos 90cos   sin90sin  sec 

2sin  cos 
2
sec 
sin 

2sin  cos 
2
sec  sec 

2
2
REF: 069040siii
3
ID: A
10 ANS:
2 sin x cos x  tan2x cos 2x
sin 2x 
sin2x
cos 2x
cos 2x
sin 2x  sin2x
REF: 069442siii
11 ANS:
1  cos A  cos 2A
cot A 
sinA  sin 2A
cos A 1  cos A  2 cos 2 A  1

sinA
sin A  2 sin A cos A
cos A cos A(1  2cos A)

sinA
sinA(1  2cos A)
cos A cos A

sinA
sinA
REF: 089442siii
12 ANS:
sin 2  sin 
 tan 
cos 2  cos   1
2 sin  cos   sin 
 tan 
2 cos 2   1  cos   1
sin (2cos   1)
 tan 
cos (2cos   1)
tan   tan 
REF: 019637siii
13 ANS:
tanx  cot x  2 csc 2x
sinx cos x
2


cos x sinx sin 2x
sin 2 x  cos 2 x
2

sinx cos x
2sinx cos x
1
1

sin x cos x sin x cos x
REF: 069640siii
4
ID: A
14 ANS:
cot x 
sin 2x
1  cos 2x
cos x
2 sin x cos x

sinx 1  (1  2 sin 2 x)
cos x 2sinx cos x

sinx
2 sin 2 x
cos x cos x

sin x
sinx
REF: 089638siii
15 ANS:
cos 2x sin 2x

 csc x
sinx
cos x
1  2sin 2 x 2 sin x cos x
1


sin x
cos x
sin x
1  2 sin 2 x
1
 2sinx 
sinx
sin x
1  2sin 2 x  2 sin 2 x
1

sinx
sin x
1
1

sinx sin x
REF: 019740siii
16 ANS:
cos A
REF: 010014siii
17 ANS:
cos 2x
 sin x  csc x  sin x
sinx
1
cos 2x
 sin x 
 sin x
sinx
sinx
cos 2x  sin 2 x  1  sin 2 x
1  2sin 2 x  sin 2 x  1  sin 2 x
1  sin 2 x  1  sin 2 x
REF: 060141siii
5
ID: A
18 ANS:
sin 
sec 

cot 
sin   cos 2
2
1

cos
sin 

2
2
2
cos 
sin   cos   sin 
sin 
sin 
1
sin 


2
cos  cos  cos 
sin 
sin 

2
cos  cos 2 
REF: 010242siii
19 ANS:
(sinx  cos x) 2  1
 (sin2x)(tanx)(csc x)
cos x
ÊÁ sinx ˆ˜ ÊÁ 1 ˆ˜
sin 2 x  2sinx cos x  cos 2 x  1
˜˜ ÁÁ
˜
 2sinx cos x ÁÁÁÁ
˜˜ ÁÁ sin x ˜˜˜
cos x
cos
x
Ë
¯Ë
¯
sin 2 x  2 sin x cos x  sin 2 x
 2sinx
cos x
2 sin x cos x
 2sinx
cos x
2sinx  2sinx
REF: 080239siii
6