10   2   Page - 1 -
Chương 

NG
 
1/
2 2 2
cos x sin x 1 2sin x- = -
. 2/
22
2cos x 1 1 2sin x- = -
.
3/

2 2 2 2
tan x sin x tan xsin x-=
. 12/
2 2 2 2
cot x cos x cot xcos x-=
.
 
1/
1
tan x cot x
sinxcosx
+=
. 2/
1 cosx sin x
sin x 1 cosx
-
=
+
.
3/
11
1
1 tan x 1 cot x
+=
++
. 4/
2
11
1 1 tan x 0
cosx cosx

7/
4
24
21
1 cot x
sin x sin x
- = -
. 8/
cosx 1
tan x
1 sin x cosx
+=
+
.
9/
( )
( )
2
1
1 cosx 1 cot x
1 cosx
- + =
+
. 10/
1 cosx 1 cosx 4cot x
1 cosx 1 cosx sin x
+-
-=
-+
.


5


 .  .
 . 
 .  .
 5.  Page - 2 - "  "
 
1/
6 6 2 2
sin x cos x 1 3sin xcos x+ = -
.
2/
( )( )
6 6 2 2 2 2
sin x cos x sin x cos x 1 sin xcos x- = - -
.
3/
( )
2
8 8 2 2 4 4
sin x cos x 1 2sin xcos x 2sin xcos x+ = - -
.
4/
( )( )
8 8 2 2 2 2

a - a a -
=
+ a a a +
. 2/
22
6
22
tan sin
tan
cot cos
a - a
=a
a - a
.
3/
22
cos sin
1 sin cos
1 tan 1 cot
aa
+ = - a a
- a - a
. 4/
( )
3
tan sin 1
sin
cos 1 cos
a - a
=

. 8/
( )
2
2
1 cos
1 cos
1 2cot
sin
sin
éù
+a
êú
-a
êú
- = a
êú
a
a
êú
ëû
.
9/
( )
( )
2
1
1 cos 1 cot
1 cos
- a + a =
+a

13/
2
2
1 sin 1 sin
4tan
1 sin 1 sin
æö
÷
+ a - a
ç
÷
ç
- = a
÷
ç
÷
ç
- a + a
÷
ç
èø
. 14/
2
2
1 cos 1 cos
4cot
1 cos 1 cos
æö
÷
+ a - a

P cos cos cot= a + a a
.
5/
( )
2 2 2
5
P 1 sin cot 1 cot= - a a + - a
. 6/
22
6
P sin tan cos cot 2sin cos= a a + a a + a a
.
10   2   Page - 3 -
7/
2
7
2cos 1
P
sin cos
a-
=
a + a
. 8/
8
2
1 cos 1
P

=-
. 12/
12
2
cosxtan x
P cot xcosx
sin x
=-
.
13/
22
13
22
cos x cot x
P
sin x tan x
-
=
-
. 14/
22
2
14
2
1 sin x cos x
P cos x
cos x
-
=-
.

. 18/
22
2
18
2
cot x cos x
P 1 cos x
cot x
-
= + -
.
19/
22
22
19
22
cos x sin y
P cot xcot y
sin xsin y
-
=-
. 20/
20
1 sin x cosxcot x
P
sin x 1 cot x 1 tan x
= - -
++
.
21/

24
22
1
P , 0 x
2
sinx cot x cos x
æö
p
÷
ç
÷
= < <
ç
÷
ç
÷
ç
èø

.
 
1/
2
A 2cos x 1=-
.
2/
2
B 3 4sin x=-
.
3/

J cos x sin x sinx cosx= - + +
.
11/
32
K cos x cos x 2sin x 2= + + -
.
12/
23
L cos x sin x cosx= + +
.
13/
( )
2
M 1 cosx cos x sinx 1 cosx= + + - +
.
 5.  Page - 4 - "  "
14/
32
N 2cos x 2cos x sin x 1= + + -
.
15/
3 3 2
O cos x sin x 2sin x 1= + + -
.
16/
( )( )
2

.
24/
( ) ( )
Y 2 tan x sinx 3 cot x cosx 5= - + - +
.
25/
( ) ( )
Z 3 cot x cosx 5 tan x sin x 2= - - - -
.
 
1/
4 4 2
A cos x sin x 2sin x= - +
.
2/
4 2 2 2
B sin x sin xcos x cos x= + +
.
3/
4 2 2 2
C cos x sin xcos x sin x= + +
.
4/
( ) ( )
4 2 4 2
D cos x 2cos x 3 sin x 2sin x 3= - + -
.
5/
6 6 4 4 2
E sin x cos x 2sin x cos x sin x= + - - +

33
I 1 cot x sin x 1 tan x cos x sinx cosx= + + + - -
.
10/
( )( )
4 4 2 2
J sin x cos x 1 tan x cot x 2= + - + +
.
11/
( ) ( )
8 8 6 6 4
K 3 sin x cos x 4 cos x 2sin x 6sin x= - + - +
.
12/
( ) ( )
4 2 4 2 2 2
L sin x 1 sin x cos x 1 cos x 5sin x cos x 1= + + + + +
.
13/
( ) ( )
2
4 4 2 2 8 8
M 2 sin x cos x sin x cos x sin x cos x= + + - +
.
10   2   Page - 5 -
 
1/

( )
2
2
2 2 2
1 cot x
1
D
cot x sin x cos x
-
=-
.
5/
62
62
1 sin x 3tan x
E
cos x cos x
-
=-
.
6/
2 2 2 2
22
tan x cos x cot x sin x
F
sin x cos x

=+
.
7/

( ) ( ) ( )
2 2 2
A f x g x h x ,= + +

( ) ( ) ( )
4 4 4
B f x g x h x ,= + +

( ) ( ) ( ) ( ) ( ) ( )
2 2 2 2 2 2
C f x .g x g x .h x h x .f x= + +
.
 Cho
a sinxsiny bcosxcosy 0-=

2 2 2 2
11
Q
a sin x bcos x a sin y bcos y
=+
++

 Cho
( )
6 6 4 4
Q sin x cos x m sin x cos x= + - +

Q




 5.  Page - 6 - "  " 




10   2   Page - 7 -
NG

 

2 3 5 7 13 3 15
, , , , , , , 2, ,
3 4 9 5 3 3 4 7 15
p p p 9p p p p p p
-
.
 
( )
rad
:

0 0 0 0 0 0 0 0 0
90 , 36 , 15 , 72 , 270 , 240 , 540 , 750 , 21
.
 

0 0 0
21 30', 75 54', 12 36'

0 0 0

90

0
120

0
150

0
180
16
p

8
p

6
p


Radian
7
4
p

12
p

5
6
p
-

2
p

2
3
p

3
4
p

7 cm

( )
10 cm


30
t.
 
( )
55 cm

( )
/40 km h


 
( )
1,75 m

( )
1,26 m

15



 5. 
= + p
.
3/
x 4k=p
. 4/
xk
3
p
= + p
.
5/
x k2
6
p
= - + p
. 6/
xk
4
p
= - + p
.
7/
k
x
3
p
=
. 8/
k2
x

=

k Î ¢

k Î ¥

1 2
 
35
3
p

m
5
p



 
1/
sinx, cosx, tan x, cot x

3
x
2
p
p < <
.
2/


5/
( )
0 0 0
C sin225 .tan130 .cot 175=-
. 6/
( )
0 0 0
D cos195 .tan269 .cot 98=-
.
7/
( )
00
E sin50 .cos 300=-
. 8/
0
21
F sin215 .tan
7
p
=
.
9/
32
G cot .sin
53
æö
pp
÷
ç
÷

.
10   2   Page - 9 -
3/
3
sin x
5
=-

3
x
2
p
p < <
. 4/
1
cosx
4
=

0x
2
p
<<
.
5/
3
cosx

00
270 x 360<<
.
9/
5
sin x
13
=

x
2
p
< < p
. 10/
1
sin x
3
=-

00
180 x 270<<
.
11/
tanx 3=

3
x
2
p
p < <

4
=

3
x
2
p
p < <
. 16/
tanx 2=-

x
2
p
< < p
.
17/
2
cot x
3
=

0x
2
p
<<
. 18/
cot x 3=-

x

cot x 2=


12
3sin x cosx sin x 3cosx
B , B
sin x cosx sin x 3cosx

==
++
.
3/ Cho
cot x 2=


12
2
2sin x 3cosx 2
C , C
3sin x 2cosx
cos x sinxcosx
+
==
-
-
.
4/ Cho
tanx 2=

1

sin x 2cos x
+
=
-

33
5
3
8cos x 2sin x cosx
D,
2cosx sin x
-+
=
-

33
6
32
sin x 2cos x 2sin x 3cosx
D
sin x cos x 5sinx
- + +
=
-+
.
5/ Cho
3
sinx , 0 x
52
p

cot x 3tan x
G
2cot x tan x
+
=
+
.
 5.  Page - 10 - "  "
8/ Cho

0
2
sinx , 0 x 90
3
= < <


12
2
tan x cosx tan x cosx
H , H cosxcot x
cot x
sin x
-
= = -
.
9/ Cho

2/
B sinx cosx=-
. :
7
B
4
=±
.
3/
33
C sin x cos x=-
. :
41 7
C
128
=±
.
 Cho
tanx cot x 3-=

1/
22
A tan x cot x=+
. :
A 11=
.
2/
B tanx cot x=+
. :
B 13=±

sinx, cosx, tan x, cot x

1/
sin x cosx 2+=
. 2/
sinx cosx 2-=
.
3/
1
sinx cosx
2
+=
.  
4/
1
sinx cosx
5
+=
. :
4 3 4 3
; ; ;
5 5 3 4
- - -
.
5/
tanx cot x 4+=
. :
1 2 3
; ; 2 3; 2 3
2

44
A sin x 3cos x=+
. :
A
7
4
=
.
10   2   Page - 11 -
2/ Cho
44
1
3sin x cos x
2
-=

44
B sin x 3cos x=+
. :
B1=
.
3/ Cho
44
7
4sin x 3cos x
4
+=

370 , 512 , 765 , 1000 , 1234-
.
 
k2a + p

k Î ¢
.

8 11 7 18 27 25
, , , , ,
3 5 2 7 10 4
p p p p p p

.
 
1/
0
sin150
. 2/
0
cot135
. 3/
0
cos225
. 4/
0
tan210
.
5/
0

3
p
. 15/
25
cot
4
p
. 16/
17
sin
3
p
.
17/
29
cos
6
p
. 18/
45
tan
4
p
. 19/
16
cos
3
æö
p
÷

æö
p
÷
ç
÷
-
ç
÷
ç
÷
ç
èø
. 22/
26
cos
3
æö
p
÷
ç
÷
-
ç
÷
ç
÷
ç
èø
. 23/
159

.
 
1/
( )
0
sin x 90-
. 2/
( )
0
cos 180 x+
. 3/
( )
0
sin 270 x-
. 4/
( )
0
sin x 180-
.
5/
( )
0
cos x 540+
. 6/
( )
0
cot 180 x+
. 7/
( )
0

. 3/
( )
tan 2 xp-
. 4/
( )
cot 3 xp-
.
5/
( )
sin x 7-p
. 6/
( )
tan x 5-p
. 7/
5
sin x
2
æö
p
÷
ç
÷
+
ç
÷
ç
÷
ç
èø
. 8/

ç
èø
. 10/
5
cos x
2
æö
p
÷
ç
÷
-
ç
÷
ç
÷
ç
èø
. 11/
11
tan x
2
æö
p
÷
ç
÷
+
ç
÷

( )
9
cos xp-
. 15/
( )
11
cot x 3-p
. 16/
( )
4
cos x 3+p
.
17/
( )
2
cot x 5-p
. 18/
( )
6
cos x -p
. 19/
8
cos x
2
æö
3p
÷
ç
÷
-

ç
÷
-
ç
÷
ç
÷
ç
èø
. 22/
2
9
tan x
2
æö
p
÷
ç
÷
-
ç
÷
ç
÷
ç
èø
. 23/
2013
7
cos x

.
25/
2015
11
cos x
2
æö
p
÷
ç
÷
-
ç
÷
ç
÷
ç
èø
. 26/
2
9
cot x
2
æö
p
÷
ç
÷
-
ç

2
æö
p
÷
ç
÷
= - + - p
ç
÷
ç
÷
ç
èø
.
2/
B cos x sin x cos x sin x
2 2 2 2
æ ö æ ö æ ö æ ö
p p p p
÷ ÷ ÷ ÷
ç ç ç ç
÷ ÷ ÷ ÷
= - + - - + - +
ç ç ç ç
÷ ÷ ÷ ÷
ç ç ç ç
÷ ÷ ÷ ÷
ç ç ç ç
è ø è ø è ø è ø
.

ç ç ç
è ø è ø è ø
.
5/
73
E 2cosx 3cos( x) 5sin x cot x
22
æ ö æ ö
pp
÷÷
çç
÷÷
= - p - + - + -
çç
÷÷
çç
è ø è ø
.
6/
( ) ( )
3
F sin 5 x cos x cot 3 x tan x
22
æ ö æ ö
pp
÷÷
çç
÷÷
= p + + - + p - + -
çç

pp
÷÷
çç
÷÷
= p + - - + p - + -
çç
÷÷
çç
÷÷
çç
è ø è ø
.
9/
( ) ( )
33
I cos 5 x sin x tan x cot 3 x
22
æ ö æ ö
pp
÷÷
çç
÷÷
= p - - + + - + p -
çç
÷÷
çç
è ø è ø
.
10/
( ) ( ) ( ) ( )

13/
( ) ( )
2013 2013 2012 2011
M cos x cos x .sin x sin x
2
æö
p
÷
ç
÷
= + p + p + - -
ç
÷
ç
÷
ç
èø
.
10   2   Page - 13 -
14/
( ) ( ) ( )
6 6 4 4 2
3
N sin x cos x 2sin x 2 sin x cos x
22
æ ö æ ö
pp

÷
ç
èø
=
æö
p
÷
ç
÷
- - p
ç
÷
ç
÷
ç
èø
.
16/
( ) ( )
22
85 3
P sin x cos 207 x sin 33 x sin x
22
æ ö æ ö
pp
÷÷
çç
÷÷
= + + p + + p + + -
çç

.
7/
0 0 0 0
G cot195 cot210 cot225 cot345= + + + +
.
8/
0 0 0 0
H cot15 .cot 35 .cot55 .cot 75=
.
9/
0 0 0 0
I tan10 .tan20 .tan30 tan80=
.
10/
0 0 0 0
J tan1.tan2 .tan3 ta n 89=
.
11/
2 0 2 0 2 0 2 0
K sin 28 sin 36 sin 54 cos 152= + + +
.
12/
2 0 2 0 2 0 2 0
L cos 2 cos 4 cos 6 cos 88= + + + +
.
13/
2 0 2 0 2 0 2 0
M sin 10 sin 20 sin 30 sin 90= + + + +
.
14/

6/
( )
00
0
00
sin 234 cos216
F .tan 36
sin144 cos126

=
-
.
 5.  Page - 14 - "  "
7/
( )
00
0
00
sin 324 cos666
G .cot 36
sin1206 cos36
-+
=
+
.
8/
( ) ( ) ( )

11/
( )
00
0 0 0
2sin2550 .cos 188
1
K
tan 368 2cos638 cos98
-
=+
+
.
12/
( )
0 0 0
00
0
cos44 tan226 .cos406
L cos72 .cot18
cos316
+
=-
.
13/
( )
0 0 0 0
00
0
tan 46 .sin 44 cot 136 .sin404
M tan 36 .tan 54

pp
÷
ç
÷
+-
ç
÷
ç
÷
ç
èø
.
16/
( ) ( )
( )
( ) ( )
( )
sin 4,8 .sin 5,7 cos 6,7 .cos 5,8
P
cot 5,2 tan 6,2
- p - p - p - p
=+
- p - p
.
 
1/
( )
sin B sin A C=+
. 2/
( )

A B 3C
sin cosC
2
++
=
. 10/
A B 2C 3C
tan cot
22
+-
=
.
11/
A 3B C
cos sinB
2
++
=-
. 12/
A 2B C 3B
cot tan
22
-+
=
.

10   2   Page - 15 -

.
4/
0 0 0 0
D sin36 .cos6 sin126 .cos84=-
.
5/
0 0 0 0
E cos112 .cos23 sin112 .sin23=-
.
6/
0 0 0 0
F sin200 .sin310 cos340 .cos50=+
.
7/
0 0 0 0 0
G cos11 .cos21 cos69 .cos79 cos10= + -
.
8/
0 0 0 0 0
H cos68 .cos78 cos22 .cos12 sin100= + -
.
9/
( ) ( )
0 0 0 0
I cos 53 .sin 337 sin307 .sin113= - - +
.
 
1/
0
15

0
1 tan15
A
1 tan15
+
=
-
. 2/
00
00
tan25 tan20
B
1 tan25 tan20
+
=
-
.
3/
0 0 0 0
0 0 0 0
sin10 cos20 sin20 cos10
C
cos17 cos13 sin17 sin13
+
=
-
. 4/
0 0 0
00
tan225 cot 81 cot 69

 .  .

 5.  Page - 16 - "  "
7/
2 0 2 0
G cos 75 sin 75=-
. 8/
2 0 2 0 2 0
H sin 20 sin 100 sin 140= + +
.
9/
2 0 0 2 0
I cos 10 cos110 cos 130= + +
.
10/
0 0 0 0 0 0
J tan20 .tan80 tan80 .tan140 tan140 .tan20= + +
.
11/
0 0 0 0 0 0
K tan10 .tan70 tan70 .tan130 tan130 .tan190= + +
.
12/
0 0 0 0 0 0
L sin160 cos110 sin250 cos340 tan110 tan340= + +
.
13/

.
2/
B sin x
3
æö
p
÷
ç
÷
=-
ç
÷
ç
÷
ç
èø

12
cosx
13
=-

3
x
2
p
p < <
.
3/
( )

< < p
.
5/
E cos x
3
æö
p
÷
ç
÷
=-
ç
÷
ç
èø

12
sinx
13
=-

3
x2
2
p
< < p
.
6/
F cot x
4

ç
÷
=+
ç
÷
ç
÷
ç
èø

5
cot x 2
2
æö
p
÷
ç
÷
-=
ç
÷
ç
÷
ç
èø
.
8/
7
H sin 2x
4

=
.
 
( )

00
4
sin , 0 90
5
a = < a <

( )

00
8
sin , 90 180
17
b = < b <


( )
A cos= a + b

( )
B sin= a - b
.
 
85
sin , tanb
17 12


 Page - 17 -
 Cho
3
p
a - b =

1/
( ) ( )
22
A cos cos sin sin= a + b + a + b
. 2/
( ) ( )
22
B cos sin cos sin= a + b + b- a
.
 
1/
A sinx 3cosx=-
. 2/
B 3sin7x cos7x=-
.
3/
( )

22
C a sin x bcosx, a b 0= + + ¹
. 4/
D 3 sin x sin x
36

.
7/
G 2sin 2x 4sin x 1
6
æö
p
÷
ç
÷
= - + +
ç
÷
ç
÷
ç
èø
. 8/
H sin2x 2 2 cosx 2sin x 3
4
æö
p
÷
ç
÷
= + + + +
ç
÷
ç
÷
ç

.
8/
H sin x cos x sin x cos x
3 4 4 3
æ ö æ ö æ ö æ ö
p p p p
÷ ÷ ÷ ÷
ç ç ç ç
÷ ÷ ÷ ÷
= - - + - -
ç ç ç ç
÷ ÷ ÷ ÷
ç ç ç ç
÷ ÷ ÷ ÷
ç ç ç ç
è ø è ø è ø è ø
.
9/
3
I cos x cos x cos x cos x
3 4 6 4
æ ö æ ö æ ö æ ö
p p p p
÷ ÷ ÷ ÷
ç ç ç ç
÷ ÷ ÷ ÷
= - + + + +
ç ç ç ç
÷ ÷ ÷ ÷
ç ç ç ç

ç ç ç ç
÷ ÷ ÷ ÷
ç ç ç ç
÷ ÷ ÷ ÷
ç ç ç ç
è ø è ø è ø è ø
.
12/
13 13 3
L cos x cos x cos x cos x
3 4 6 4
æ ö æ ö æ ö æ ö
p p p p
÷ ÷ ÷ ÷
ç ç ç ç
÷ ÷ ÷ ÷
= - + + + +
ç ç ç ç
÷ ÷ ÷ ÷
ç ç ç ç
÷ ÷ ÷ ÷
ç ç ç ç
è ø è ø è ø è ø
.
 
1/
tan 3x tan x
A
1 tan x tan 3x
-

1 tan 2xtan x
-
=
-
.
 
1/
( ) ( )
A sin a b sin a sin b
2
æö
p
÷
ç
÷
= + + - -
ç
÷
ç
÷
ç
èø
.
2/
( )
B cos a cos b cos a b
22
æ ö æ ö
pp
÷÷

D sin a sin b cos a cos b=-
.
5/
( )
( ) ( )
2sin a b
E tan b
cos a b cos a b
+
=-
+ + -
.
 
1/
22
A cos x 3sin2x sin x= - -
. 2/
3
B 4sin x 3sinx 3cos3x= - +
.
3/
( ) ( )
00
C sin x 45 cos x 45= + + +
. 4/
D tan3x tanx sin2x= - -
.
5/
2
E tan2x cot x 8cos x= + -

÷÷
çç
÷÷
+ = - = +
çç
÷÷
çç
÷÷
çç
è ø è ø
.
7/
cosx sin x 2 cos x 2 sin x
44
æ ö æ ö
pp
÷÷
çç
÷÷
- = + = - -
çç
÷÷
çç
÷÷
çç
è ø è ø
.
8/
( ) ( )
2 2 2 2

4sin x sin x 4sin x 3
33
æ ö æ ö
pp
÷÷
çç
÷÷
+ - = -
çç
÷÷
çç
÷÷
çç
è ø è ø
.
3/
( ) ( ) ( )
sin xsin y z sin ysin z x sinzsin x y 0- + - + - =
.
4/
( ) ( ) ( )
cosxsin y z cosysin z x coszsin x y 0- + - + - =
.
5/
( ) ( )
tan x y tan x tan y tan x y tan xtan y+ - - = +
.
6/
ta n2x tan x tan2xtan x tan x tan x 1
6 3 3 6

32
cos x .cos x cos x .cos x 1 3
3 4 6 4 4
æ ö æ ö æ ö æ ö
p p p p
÷ ÷ ÷ ÷
ç ç ç ç
÷ ÷ ÷ ÷
- + + + + = -
ç ç ç ç
÷ ÷ ÷ ÷
ç ç ç ç
è ø è ø è ø è ø
.
9/
( )( ) ( )( )
o o o o o o o o
cos70 cos50 cos230 cos290 cos40 cos160 cos320 cos380 0+ + + + + =
.
 
1/
( )
( )
cos a b
cot a cot b 1
cot a cot b 1
cos a b
-
+
=

.
 Chng minh các h thc sau, vu kic
1/ 
( )
cos a b 0+=

( )
sin a 2b sina+=
.
2/ 
( )
sin 2a b 3sinb+=

( )
tan a b 2tana+=
. HD:
( )
( )
b a b a
2a b a b a
ì
ï
= + -
ï
ï
í
ï
+ = + +
ï
ï


( )
2tana tan a b=+
. HD:
( )
b a b a= + -
.
 5.  Page - 20 - "  "
7/ 
( )
cos 2a b 1+=

( )
b
tan a b tana 2tan
2
+ - =
.
8/ 
( ) ( )
cos a b kcos a b+ = -

1k
tan a tan b
1k
-
=

cos x cos x y cos y 2cosxcosycos x y 1+ + + - + =
.
 
 
1/
2
A sin x cos x cos x
33
æ ö æ ö
pp
÷÷
çç
÷÷
= + - +
çç
÷÷
çç
÷÷
çç
è ø è ø
.
2/
2 2 2
B cos x cos x cos x
33
æ ö æ ö
pp
÷÷
çç
÷÷

33
æ ö æ ö
pp
÷÷
çç
÷÷
= + + + -
çç
÷÷
çç
÷÷
çç
è ø è ø
.
5/

33
a cos x cos3x a sin x sin3x
F , a const
cosx sin x
-+
= + =
.
 
tan x tan x tan x tan 3x
33
æ ö æ ö
pp
÷÷
çç

cosA.cosB
= + ¹
.
5/
( )

0
tan A tan B tanC tanA.tanB.tanC, A,B,C 90+ + = ¹
.
6/
cot A.cot B cot B.cotC cotC.cot A 1+ + =
.
7/
A B C B C
sin cos cos sin sin
2 2 2 2 2
=-
.
8/
A B C B C
cos sin cos cos sin
2 2 2 2 2
=-
.
10   2   Page - 21 -
9/
A B B C C A

+ + = +
.
14/
sinAcosBcosC sinBcosCcosA sinCcosAcosB sinAsinBsinC+ + =
.
 
A,B,C
ng minh
1/
tanA tanB tanC 3 3, A,B,C+ + ³ "

2/

2 2 2
tan A tan B tan C 9, A,B,C+ + ³ "

3/

6 6 6
tan A tan B tan C 81, A,B,C+ + ³ "

4/
2 2 2
A B C
tan ta n tan 1
2 2 2
+ + ³
.
5/
A B C

++
ç
÷
ç
èø

3/
.  5.  Page - 22 - "  "


. 6/
0
20
tan15
F
1 tan 15
=
-
.
7/
2
7
tan
8
G
1 tan
8
p
=
p
-
. 8/
00
13
H
sin10 cos10
=-
.
9/
0

 .
 .
 .
 
 .  .
 .  .
 .  .
10   2   Page - 23 -
 
1/
00
A cos36 cos72=
. 2/
2
B cos cos
55
pp
=
.
3/
C sin .cos cos
8 8 4
p p p
=
. 4/
D sin .cos .cos
16 16 8

11/
0 0 0 0 0
K cos10 .cos20 .cos30 cos70 .cos80=
.
12/
0 0 0 0
L 8tan18 cos18 cos36 cos72=
.
13/
0 0 0 0
M cos20 cos40 cos60 cos80=
.
14/
234
N cos cos cos cos
9 9 9 9
p p p p
=
.
15/
O 96 3sin cos cos cos cos
48 48 24 12 6
p p p p p
=
.
16/
2 4 8 16 32
P cos .cos .cos .cos .cos
31 31 31 31 31
p p p p p

4 0 4 0
sin60
B 3sin15 cos15
sin 15 cos 15
=-
-
.
3/
0 0 0 0
C ta n9 tan27 tan63 tan81= - - +
.
4/
D 4tan 2tan tan cot
8 16 32 32
p p p p
= + + -
.

5/
( )
2 0 0 0 2 0
2 0 2 0
cos 696 tan 260 tan530 cos 156
E
tan 252 cot 342
+ - -
=
+
.
 

çç
÷÷
çç
è ø è ø
. 6/
F sinxcosxcos2x=
.
7/
G 4sin xsin x sin 2x
22
æ ö æ ö
pp
÷÷
çç
÷÷
= + +
çç
÷÷
çç
÷÷
çç
è ø è ø
. 8/
22
xx
H sin sin
8 2 8 2
æ ö æ ö
pp
÷÷

B
cos2x sin2x
-
=
-
.
3/
( )( )
2
C
1 tan x 1 cot x
=
-+
. 4/
( )
2
D 1 tan x cot x=-
.
5/
sin2x cos2x
E
sinx cosx
=-
. 6/
cot x tan x
F
cos2x
-
=
.

ç
èø
. 8/
22
22
sin 2x 4sin x
H
sin 2x 4sin x 4
-
=
+-
.
9/
sin 4x cos2x
I.
1 cos4x 1 cos2x
=
++
. 10/
sin 3x cos3x
J
sin x cosx
-
=
+
.
11/
1 cos x
2
x

÷
ç
÷
+
ç
÷
ç
÷
ç
èø
. 12/
1
L cot2x tanx
2sin2x
= + +
.
13/
1 cos2x sin2x
M .cot x
1 cos2x sin2x
-+
=
++
. 14/
( )
2
2 sin2x 2cos x 1
N
cosx sin x cos3x sin3x
+-

ç
÷
= - + + - < <
ç
÷
ç
÷
ç
èø
.
 Tính giá tr ca biu thng giác, khi bit
1/

3
sin2x, cos2x khi sinx , x
52
p
= < < p
.
2/
sin2x, cos2x khi sinx cosx 2+=
.
10   2   Page - 25 -
3/
53
cos2x, sin2x, tan2x khi cosx , x
13 2


x
16
p
=
.
2/
4 3 3 4
B sin x sin xcosx cos xsin x cos x= + - +

x
48
p
=
.
3/
C cos4x=

tanx cot x 3+=
.
4/
( )
0
D cos 270 4x=+

( )
0
cot 45 x 2+=
.
5/

èø
.
 
0
51
sin18
4
-
=
.
 
1/
0
cos15
. 2/
0
cos18
. 3/
0
sin36
. 4/
0
cos36
.
 
x
tan
2

1/

ç
÷
= < <
ç
÷
ç
÷
ç
èø


1
ta n y , 0 y
32
æö
p
÷
ç
÷
= < <
ç
÷
ç
÷
ç
èø
.

xy+
. 2/

4
ì
ï
p
ï
<<
ï
ï
ï
í
ï
+
ï
=
ï
ï
ï
î
.