)

Page - 1 -


 

22
sin x cos x 1

tanx.cotx 1

sin x
tan x
cos x


cos x
cotx
sin x

os
2
2
1
1 tan x
cx

2
2
1

sin3x 3sinx 4sin x

3
cos3x 4 cos x 3cosx

 

sin a b sina.cosb cosa.sinb

osc a b cosa.cosb sina.sin b


tana tan b
tan a b
1 tana.tan b

tana tan b
tan a b
1 tana.tan b


π 1 tan x
tan x
4 1 tan x

π 1 tan x
tan x
4 1 tan x

 

cosa.cosb
2

sin a b sin a b
sina.cosb
2


cos a b cos a b
sina.sin b
2

 

ππ
sin x cos x 2 sin x 2 cos x
44

ππ
sin x cos x 2 sin x 2 cos x
44


4 4 2
1 cos 4x
cos x sin x 1 s
31
in 2x
2 4


.
 
cotx

sin x 0 x k k
.
 
tanx

cotx

x k. k
2
.
 

 
x


 

"
AM


k2
n

7
6

k 0,k 1
).

2
AM k.
43

M

11
;
4 12

19
12
,
k 0;1;2
.

k2
AM k.
4 2 4 4

M

4

xk
3
 
3

4
3
.


xk
32

 
2
2
11
cos x cos x
22
11
sin x sin x
22




/3
5/6
4/3
/6

cos x 1 sin x 0

 '' c s p''
 ''''

 
cos cos

''  ''
sin sin , tan tan , cot tan

 
sin sin

''  ''
cos cos , tan tan , cot tan

 
0


sin cos , cos sin , tan cot , cot tan
2 2 2 2

  '' '' 

sinu cosv


sinu sinv


Ths.   (Nâng cao) Page - 4 -

 :
u v k2
sin u sin v
u v k2
:
sin x 0 x k
sin x 1 x k2
2
sin x 1 x k2
2

 :
u v k2
cos u cos v
u v k2
:
cos x 0 x k
2
cos x 1 x k2
cos x 1 x k2

 :
tan u tan v u v k
Ðk : u,v k

sin x 1 cos2x sin2x 1 cos x



1 1 7
4 sin x
sin x 4
3
sin x
2



44
7
sin x cos x cot x cot x
8 3 6



44
4
sin 2x cos 2x
cos 4x
tan x tan x
44

9

3 x 1 3x




2 2 2
3
sin x sin 2x sin 3x
2
.


2 2 2
sin x sin 2x sin 3x 2
.


2 2 2 2
sin x sin 3x cos 2x cos 4x



2 2 2 2
sin 3x cos 4x sin 5x cos 6x


sin
22
5x 9x
cos 3x sin7x 2 2cos
4 2 2


2
2sin x 1 3cos4x 2sin x 4 4cos x 3



6 6 8 8
sin x cos x 2 sin x cos x



8 8 10 10
5
sin x cos x 2 sin x cos x cos2x
4



3 3 5 5
sin x cos x 2 sin x cos x



4 2 2 4
3cos x 4cos x sin x sin x 0



33
2 3 2
cos 3x cos x sin 3x sin x


2
1 sin2x cos2x
2 sin x sin2x
1 cot x


tan x cotx 2 sin2x cos2x



2
tan x tan x tan3x 2



222
11
tan x cot x cot 2x
3



2 2 2
xx
sin tan x cos 0
2 4 2




4 1 sin x


cos3x tan5x sin7x



11
sin2x sin x 2cotx
2sin x sin2x



44
sin x cos x 1
tan x cot2x
sin2x 2



2 2 2 2
tan x.cot 2x.cot3x tan x cot 2x cot3x



x
cotx sin x 1 tan x tan 4
2
Câu 7. 
1 cos 4x sin 4x
0
2sin2x 1 cos4x
.
Câu 8. 
2
21
sin x cos x cos x
2
.
Câu 9. 
2
x
2 3 cos x 2sin
24
1
2cos x 1
.
Câu 10. 
sin4x 3sin2x tanx
.
Câu 11. 
23
cos10x 2cos 4x 6cos3x cosx cosx 8cosx cos 3x
.
Câu 12. 
2 2 2
2cos x 2cos 2x 2cos 3x 3 cos4x 2sin2x 1
.

3
.
Câu 19. 
11
2 2 sin x
4 sin x cos x
.
Câu 20. 
2
2 tan x cot2x 3
sin2x
.
Câu 21. 
2
3 tan 3x cot2x 2 tan x
sin4 x
.
Câu 22. 
2 2 2
sin x sin 2x sin 3x 2
.
Câu 23. 
25 4x 3 sin2 x 8sin x 0
.
Câu 24. 
sin2x
2cos x 0
1 sin2x
.
Câu 25. 

.
Câu 30. 
2
2sin 3x 1 4 sin x 1
.
Câu 31. 
3 3 2
cos x 4sin x 3cosx sin x sinx 0
.
Câu 32. 
44
xx
sin cos 1 2 sin x
22
.
Câu 33. 
sin 3x sin2x sin x
44
.
Câu 34. 
2
4
4
2 sin x sin3x
tan x 1
cos x
.
Câu 35. 
2
x

9sinx 6cos x 3sin2x cos2x 8
.
Câu 43. 
cosx sin x cos x sin x cos x cos2x
.
Câu 44. 
2
2sin x 1 3cos4x 2sin x 4 4cos x 3
.
Câu 45. 
33
2sin x sinx 2cos x cosx cos2x
.
Câu 46. 
2 3 4 2 3 4
sinx sin x sin x sin x cosx cos x cos x cos x
.
Câu 47. 
2
sin x tan x 1 3sin x cosx sin x 3
.
Câu 48. 
2
tan2x cotx 8cos x
.
Câu 49. 
3 cotx cos x 5 tan x sin x 2
.
Câu 50. 
11


2
a tan x btanx c 0t tanx

x k , (k )
2

2
a cot x bcotx c 0t cotx

x k , k

t
2
t sin x
hoc
t sin x
u kin là
0 t 1


cos
)


cos x sin x sin x cos x sin2x


22
cos x sin x cos x sin x 2cos2x
cotx tan x 2cotx
sin x cos x sin x cos x sin2x


4 4 2 2
1 1 1 3 1cos 4x
sin x cos x 1 sin 2x cos 2x
2 2 2 4


4 4 2 2 2 2
cos x sin x sin x cos x cos x sin x cos2x


6 6 4 4 2 2 2
3 5 3cos4x
sin x cos x sin x cos x sin x cos x 1 sin 2x
48


6 6 4 4 2 2
cos x sin x cos2x sin x cos x sin x cos x


x1


cos 3x sin3x
5 sin x 3 cos2x , x 0;2
1 2sin2x



sin 3x sin5x
35



sin5x
1
5sin x



22
cos 3xcos2x cos x 0 1



44
3
cos x sin x cos x sin 3x 0
4 4 2







11
2sin 3x 2cos 3x
sin x cos x



2
cos x 2sin x 3 2 2cos x 1
1
1 sin2x



x 3x x 3x 1
cos x cos cos sin x sin sin
2 2 2 2 2



44
sin x cos x 1 1
cot2x
5sin2x 2 8sin2x



22

1
sin x cos x
8



8 8 2
17
sin x cos x cos 2x
16



3
5x x
sin 5cos x sin
22



2
sin2x cotx tan2x 4 cos x



2
6x 8x
2cos 1 3cos
55



2
cos2x 1
cotx 1 sin x sin2x
1 tan x 2


sin2x 2tan x 3


1 tan x 1 sin2x 1 tan x



2
cos2x cosx 2 tan x 1 2


3
sin2x cos x 3 2 3 cos x 3 3 cos2x 8 3 cos x sin x 3 3


2
2
11
4 sin x 4 sin x 7 0
sin x
sin x
2
2
11
cos x 2 cos x 1
cos x
cos x
1
2 tan x cotx 2sin2x
sin2x
22
tan x cot x 2 1 tan x cotx 0Ths.   (Nâng cao) Page - 12 -


44
11
sin x cos x sin2x
8




Câu 51. 
3
4cos x 3 2 sin2x 8 cos x

Câu 52. 
2
6sin 3x cos12x 14

Câu 53. 
3 tan x cotx 1 3

Câu 54. 
tan x 3 cotx 1 3

Câu 55. 
22
13
4
sin x cos x
sin x cos x

Câu 56. 
2
1
4 tan x 2 0
cos x

Câu 57. 

4 4 8

Câu 63. 
tanx cotx 4

Câu 64. 
2
sin x 3 2 2cos x 2sin x 1
1
1 sin2x

Câu 65. 
4
4cos x 3 2 sin2x 8cosx

Câu 66. 
1 1 2
cos x sin2x sin 4x

Câu 67. 
sin2x 2 sin x 1
4

)

Page - 13 -
Câu 68. :
2 2sin x 1 4 sin x 1 cos 2x sin 2x
44


2
3cos4x 2cos 3x 1

Câu 76. 
x
cos x tan 1
2

Câu 77. 
2
3tan2x 4 tan3x tan 3x tan2x

Câu 78. 
2
3
cos x cos4x cos2x cos3x cos 4x
2

Câu 79. 
1 tan x 1 sin2x 1 tanx

Câu 80. 
6 6 2
13
sin x cos x cos 2x
8

Câu 81. 
66
1

Câu 87. 
2cosxcos2x 1 cos2x cos3x

Câu 88. 
sin3x cos2x 2 sin2x cos x 1

Câu 89. 
44
2 cos 2x sin 2x cos8x cos4x 0

Câu 90. 
2
x 5 17 1 9
2cos 2 2x cos 10cos x sin x
2 2 2 2 2

Câu 91. 
5
4 cos x cos 2x
3 3 2

Câu 92. 
44
5
sin x cos x 2sin x
2

Câu 93. 
66
1

44
4
xx
sin cos
22
cos x
tan x tan x
44

Câu 99. 
2
2
11
cos x cos x
cos x
cos x

Câu 100. 
cos2x 5 2 2 cosx sin x cosx

)

Page - 15 -


:
a sin x bcos x c , a,b \ 0

 :
 


 
xx
cos 0 k x k2
2 2 2
 2
22
x 2t 1 t
t tan sin x , cos x
2
1 t 1 t


2
(b c)t 2at c b 0
.
 
x k2 b c 0
nên 
2 2 2 2 2 2
' a (c b ) 0 a b c
.
  
t

x
tan t x

Ths.   (Nâng cao) Page - 16 -


31
8 sin x
cos x sin x



3
sin x cosx sin2x 3 cos3x 2 cos4x sin x



3
3sin3x 3 cos9x 1 4 sin 3x


3 cos5x 2sin3x cos2x sin x 0


9sin x 6cosx 3sin2x cos2x 8


sin2x 2 cos2x 1 sin x 4cosx



tan x 3cotx 4 sin x 3 cos x

. 

33
sin x cos x sin x cos x



44
1
cos x sin x
44



33
4 sin x cos3x 4cos x sin3x 3 3 cos4x 3


2 2 sin x cos x cosx 3 cos2x
2cos x 1 sin x cos x 1
2cos2x 6 cosx sin x



2
tan x sin2x cos2x 4 cos x
cos x)

Page - 17 -

2
x
2 3 cos x 2sin
24
1
2cos x 1
2
3cos x 4sin x 6 2 3 3 cos x 4sin x 6

sin x sin2x
3
cos x cos2x




cos3x sin x 3 cos x sin3x



3
4 sin2x 3cos2x 5cos 3x 0
2



22
x3
4 sin 3 cos2x 1 2cos x , x 0;
24


cos2x 3 sin2x 3 sin x cos x 4 0



3
sin x cos x 2 sin2x 1 sin x cos x 2



22
3 cos x 2 sin x cos x 3 sin x 1 0

nh:

6

Ths.   (Nâng cao) Page - 18 -
Câu 104. 
2
21
sin 2x 3 sin 2x cos2x 2 sin x
2

Câu 105. 
32
2sin x sin x
4 4 2

Câu 106. 
cos x 3 sin x 2cos x
3

Câu 107. 
sin x 2 sin5x cos x

Câu 108. 
sin x cosx 2 2 sin x cos x

Câu 109. 
sin5x cos5x 2 cos13x


Câu 117. 
1
4 sin x 3cos x 4 1 tan x
cos x

Câu 118. 
sin5x 3 cos5x 2sin7x

Câu 119. 
3sinx cosx 1

Câu 120. 
sin x 5cosx 1

Câu 121. 
1 3 sin x 1 3 cos x 2 , x 0;

Câu 122. 

13
sin 3x 3 2 cos3x 1 , x ;
99

Câu 123. 
3 2 sin x 3 2 cos x 20

Câu 124. 
sin x 1 sin x cos x 1 cosx

Câu 125. 

Câu 132. 
2
1 cos x cos2x cos3x 2
3 3 sin x
3
2cos x cos x 1

Câu 133. 
cos2x 3 sin2x 3 sin x cos x 4 0

Câu 134. 
3
3sin2x 3 cos6x 1 4sin 2x

Câu 135. 
3
cos x 3 sin x 3
3 sin x cos x 1

Câu 136. 
cos9x 2cos6x 2 0

Câu 137. 
sin x sin2x
3
cos x cos2x

Câu 138. 
2
2cos x 3 sin2x 1 3 sin x 3 cos x

2sin x 1

Câu 146. 
3
sin2x cos x 3 2 3 cos x 3 3 cos2x 8 3 cos x sin x 3 3

Câu 147. 
22
sin x 4sin x 3 sin2x 3cos x 2 1 2sin x sin x 3 cos x

Câu 148. 
22
cos x 3 sin2x 1 sin x

Câu 149. 
23
cos x 3 sin2x sin x 1

Câu 150. 
sin3x 3 cos3x sin2x 3 cos2x sinx 3 cosx

Ths.   (Nâng cao) Page - 20 -


 .

22

2
cos X
(hay
2
sin X

22
2 2 2 2
sin X sin X cos X cos X d
1 a. b. c.
cos X cos X cos X cos X22
a tan X b tan X c d 1 tan X2
a d tan X b tan X c d 0
.
 
t tanX

 
 

22
1 cos2X 1 cos2X
sin X ; cos X
22

sin X

3
cho

4
cos X
(hay
4
sin X
)



22
cos x 3 sin2x 1 sin x



3 3 2
cos x 4sin x 3cosx sin x sin x 0



4 2 2 4
3cos x 4sin x cos x sin x 0


sin2x 2tan x 3



22
tan x sin x 2sin x 3 cos2x sin x cos x



32
cos x sin x 3sin x cos x 0



2
sin x tan x 1 3sin x cos x sinx 3


22
9
cos 3 2x 3 cos 4x 1 sin 2x , x ;2 1
23



3
2
3
1 cos x
tan x
1 sin x






33
4cos x 2sin x 3sin x 0



3
6sin x 2cos x 5sin2x cos x


1 3 tan x 2sin2x



3 3 5 5
sin x cos x 2 sin x cos x



44
3sin x 5cos x 3 0



sin x cos x
1
sin2x



Page - 22 -


3
2 2 cos x 3cos x sin x 0
4



33
4 sin x cos x cos x 3sin x



22
5
4 3 sin x cos x 4 cos x 2sin x
2



2
sin x 3sin x cos x 1 0



4 4 2
1
sin x cos x 2 sin2x cos2x cos 2x

2

22
53
3sin 3 x 2 sin x cos x 5 sin x 0
2 2 2



1
3 sin x cos x
cos x



1
4 sin x 6 cos x
cos x



3
22
7 sin x 2sin2x 3cos x 3 15 0



3 3 2
4sin x 3cos x 3sinx sin x cos x 0



Câu 154. 
3 3 2
4sin x 3cos x 3sin x sin x cos x 0

Câu 155. 
22
97
sin x 3sin x cos x 4 cos 2 x 0
22

)

Page - 23 -
Câu 156. 

22
3
2cos x 3 3 sin x cos x cos 3 x 2 , x 0;
22

Câu 157. 

22
cos 2x 3 sin4x 1 sin 2x , x ;2
3

Câu 158. 

33


Câu 166. 
7
sin 3x cos3x 2sin x 0
2

Câu 167. 
3 sin2x 2 cos x 1 2 cos3x cos2x 3cos x

Câu 168. 
3
2cos x sin3x

Câu 169. 
33
1
1 sin 2x cos 2x sin4x
2

Câu 170. 
66
8 sin x cos x 3 3 sin4x 3 3 cos2x 9sin2x 11

Câu 171. 
3 3 2
4sin x 3cos x 3sin x sin x cosx 0
E

2
. 2 2 2
PP : t tan x cotx , t 2 tan x cot x t 2
.
Ths.   (Nâng cao) Page - 24 -
 .
22
a tan x cot x b tan x cotx c 0sin x 0
k
ÐK : sin2x 0 x , k
cos x 0
2
. 2 2 2
PP : t tan x cotx , t 2 tan x cot x t 2
.
 .
44
a sin x cos x bsin2x c 0

 .
66
a sin x cos x bcos2x c 0
6 6 2 2 2
3 1 3 1 3
PP : t cos2x, t 1 sin x cos x 1 sin 2x cos 2x t
4 4 4 4 4
.
 .
44
a sin x bcos x ccos2x d 0
2
24
2
2
4
1t
1 cos2x 1 t
sin x sin x
2 2 4
PP : t cos2x, t 1
1 cos2x 1 t
1t
cos x

42
cos x

206:

33
2sin x sin x 2cos x cos x cos2x

207

2 3 4 2 3 4
sin x sin x sin x sin x cosx cos x cos x cos x

208

2 3 3
tan x 1 sin x cos x 1 0

209

3
2
cos 2x 2 sin x cosx 3sin2x 3

)

Page - 25 -
210
2sin x cotx 2sin2x 1


2
2
2
2 tan x 5 tan x 5 cotx 4 0
sin x

218

2
2
15
cot x tan x cotx 2 0
2
cos x219

33
1 cos x sin x sin x

220

32
cos x cos x 2sin x 2 0

221
cos2x 5 2 2 cos x sin x cosx

222

1 sin x cos x sin2x 2cos2x 0

230

22
sin x cos x cos2x sin x cos x sin x cosx

231
cos2x 5 2 2 cos x sin x cosx

232

23
cos x sin x cos x 0

233

3
4 sin x 1 3sin x 3 cos3x

234

22
1 sin x cos x 1 cos x sin x 1 sin2x

235
sin x cos x 2sin x 2cosx 2

236
1 tan x 2 2 sin x