Phương trình lượng giác và ứng dụng (Nâng cao)

  !
"


#$%&'())))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))*
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))+
,
-
..))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))+
/0123
-
..)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))4
,
-
.567
-
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))*8




 !"#$

#)))))))))))))))))))))))))))9*
,
-
..))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))99
/0123
-
..)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))9+
,

-
.567
-
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))+:
#

%&''))))))))))))))))))))))))))))))))))))))))))))))))))))))))))+8
,
-
..))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))+8
/0123
-
..))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))):*
,
-
.567
-
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))4;
( )%*)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))4<
,
-
..))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))4+
/0123
-
..)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))4=
,
-
.567
-
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))8*

+ !,-)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))89
,
-
..))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))8<
,
-
.567
-
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))8:
.-/-0123

 !)))))))))))))))))))))))))))))))))))8=
,
-
..))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))8=
,
-
.567
-
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))88
45$#61#7))))))))))))))))))))))))))))))))))))))))))))))))))));>;
,
-
..))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))));>*
,
-
.567
-
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))));><
8-#'!8* ))))))))));>:

,
-
..))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))));>:
,
-
.567
-
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))));;*
983

*))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))));;:
,
-
..))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))));;=
:83

-#;

<=#)))))))))))))))))))))))))))))))))))))));*;
,
-
..))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))));**
,
-
.567
-
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))));*+
?#%@3@$'AAAA?  B;
 !" 
>?;48@AB;449>;CDE;4


$-
C
2 2
sin x cos x 1+ =
C
tanx.cotx 1=
C
sinx
tanx
cosx
=
C
cosx
cot x
sinx
=
C

2
2
1
1 tan x
c x
+ =
C
2
2
1
1 cot x

sin x
+ =
$-1 $$-=

$-1
C
sin2x 2sinx.cosx=
C
2 2
2 2
cos x sin x
cos2x
2cos x 1 1 2sin x
é
-
ê
=
ê
- = -
ê
ë
C

2
1 c 2x
sin x
2
-
=
C



2
1 c 2x
c x
2
+
=
C
3
sin3x 3sinx 4sin x= -
C
3
cos3x 4cos x 3cosx= -
$-$

1
C
( )
sin a b sina.cosb cosa.sinb± = ±
C
( )
c a b cosa.cosb sina.sinb± = m
C
( )
tana tanb
tan a b
1 tana.tanb
+
+ =

-
C
( )
tana tanb
tan a b
1 tana.tanb
-
- =
+
C
D 1 tanx
tan x
4 1 tanx
æ ö
+
÷
ç
÷
+ =
ç
÷
ç
÷
ç
-
è ø
C
D 1 tanx
tan x
4 1 tanx

æ ö
-
÷
ç
÷
- =
ç
÷
ç
÷
ç
+
è ø
$-F GGH
C
a b a b
cosa cosb 2cos .cos
2 2
+ -
+ =
C
a b a b
cosa cosb 2sin .sin
2 2
+ -
- = -
C
a b a b
sina sinb 2sin .cos
2 2

+ -
+ =
C
a b a b
sina sinb 2cos .sin
2 2
+ -
- =
C
( )
sin a b
tana tanb
cosa.cosb
+
+ =
C
( )
sin a b
tana tanb
cosa.cosb
-
- =
$-F GHG
C
( ) ( )
cos a b cos a b
cosa.cosb
2
+ + -
=

C
( ) ( )
sin a b sin a b
sina.cosb
2
+ + -
=
C
( ) ( )
cos a b cos a b
sina.sinb
2
- - +
=
$

'!$-$<I5
C
D D
sinx cosx 2sin x 2cos x
4 4
æ ö æ ö
÷ ÷
ç ç
÷ ÷
+ = + = -
ç ç
÷ ÷
ç ç
÷ ÷

ç ç
è ø è ø
C
D D
sinx cosx 2sin x 2cos x
4 4
æ ö æ ö
÷ ÷
ç ç
÷ ÷
- = - = +
ç ç
÷ ÷
ç ç
÷ ÷
ç ç
è ø è ø
B* ?EBFGBF'GBBBBHF7AA?
!
"
C
4 4 2
1 cos4x
cos x sin x 1 s
3 1
in 2x
2 4
+
+ = - =
C

6 6 2
3 cos4x
cos x sin x 1 s
5 3
in 2x
4 8
+
+ = - =
$

'!1JI

"J6H
-
K
-
'L.
sinx
cosx
ộ
= a
ờ
ờ
= a
ờ
ở
I
1 1- Ê a Ê

M2.K'N
tan
!
-

cot
OK'16N!
-
!3
-
PQR
.2S!
-
SJ6H
-
ST.USV


tanx
OSJ6H
-
I
( )
cosx 0 x k k
2
p
ạ ạ + p ẻ Â

cot x
OSJ6H
-
I
( )
sinx 0 x k kạ ạ p ẻ Â


2
tanx

cot x
OSJ6H
-
I
( )
x k. k
2
p
ạ ẻ Â


M'S
-
'.2HT'0SJ6H
-
W@'$
-
6S7
STHT'SJ6H

-
I

MT'XR.3Y7VL
x
3T6SJ6H
-
R6HROV
Q7'SZS[
-

-
'OR6\
-
'

]@SW^O_3T6`a6LSJ6H
-
6L

-
'R6a67@6\
-
'OR6H$@
-


-
'
#3T6`6bKSW^I?R66!

-
K
ẳ
AM
K
NS
k2
n
p
a +


0
0
k.360
hay a
n
ổ ử
ữ
ỗ
ữ
+
ỗ
ữ
ỗ
ữ
ỗ
ố ứ
0
k ,n

+
ẻ ẻÂ Ơ
K
n
ST'
M
SW^
SJ66?
c;IR6S
ẳ
AM k2
3
p
= + p
K'$
-
ST'
M
\Vc
3
p
a
k 0=

c*IR6S
ẳ
AM k
6
p
= + p

K*ST'
M
\Vc
6
p

7
6
p
a
k 0,k 1= =

c9IR6S
ẳ
2
AM k.
4 3
p p
= +
K9ST'
M
\Vc
11
;
4 12
p p

19
12
p

O
( )
k 0;1;2=

?#%@3@$'AAAA? B9

!"#$%
$&'(((&
)%)*"+,
-
 !" 
c<IR6S
¼
k2
AM k.
4 2 4 4
p p p p
= + = +
K<ST'
M
\Vc
4
p
O
3
4
p
O
5
4

p
d
7
4
p

0Vc
k 0,1,2,3=

c+Ie.6
x k
6
p
= - + p

x k
3
p
= + p
,T6`6
x k
6
p
= - + p
SW^K*ST'\VcI
6
p
-

5

6
p
,T6`6
x k
3
p
= + p
SW^K
*ST'\VcI
3
p

4
3
p

e.6f'<ST'g
6e.I
x k
3 2
p p
= +

"N0.
2
2
1 1
cos x cosx
2 2
1 1

sin x sinx
2 2
é é
ê ê
= = ±
ê ê
Û
ê ê
ê ê
= = ±
ê ê
ë ë

H$2
XR.HSKK0<
-
'OHHR.0SJ6H
-
Q.\.O\3
-

N6Q_I
2
2
2
2
1
cos x
2cos x 1 0 cos2x 0
2

1 cos2x 0
2sin x 1 0
sin x
2
é
ê
é
=
é
- = =
ê
ê
ê
Û Û
ê
ê
ê
=
- =
ê
ê
ê
ë
=
ë
ê
ë
XSN0.
2
2

sin x 1 sinx 1
cosx 1
cos x 1
é
é
= = ±
ê
ê
Û
ê
ê
= ±
=
ê
ê
ë
ë
H$2RO'3RSeX$
2 2
sin x cos x 1+ =
[SKI
2 2
2 2
sin x 1 cos x 0 cosx 0
sinx 0
cos x 1 sin x 0
é é
é
= = =
ê ê

ê
Û Û
ê ê
ê
=
= =
ê ê
ê
ë
ë ë

hi\6%[IKK' !LMINKK

"7KTUB'6%[jjS2O`0jjK\SK^
'$
-
(N%RO
-
6k62QH2.

' !O_LKSN63Y6O
( )
cos cos- a = a
O^6
K^\3YjjbjjcKI
( ) ( ) ( )
 sin sin , tan tan , cot tan- a = - a - a = - a - a = - a

LMO_LK3@63Y6O
( )

sin sinp- a = a
O^6
K^\3YjjbjjcKI
( ) ( ) ( )
 cos cos , tan tan , cot tanp- a = - a p- a =- a p- a =- a
B< ?EBFGBF'GBBBBHF7AA?
l9
+l:
<l9
bl:
m
!
"

INO_0K.6Ke3Y8>
>
K73YKH
\OI
sin cos , cos sin , tan cot , cot tan
2 2 2 2
ổ ử ổ ử ổ ử ổ ử
p p p p
ữ ữ ữ ữ
ỗ ỗ ỗ ỗ
ữ ữ ữ ữ
- a = a - a = a - a = a - a = a
ỗ ỗ ỗ ỗ
ữ ữ ữ ữ
ỗ ỗ ỗ ỗ
ữ ữ ữ ữ

ỗ ỗ ỗ ỗ
ố ứ ố ứ ố ứ ố ứ

n7iSR0co6S7STQ7S
-
6k62Ljj6%[jj7I
p2.I
sinu cosv=
qrOs.%.32Ot3R2.
sinu sinv=
O
-
7^.
sinu cosv=
u
#62WsS7c.vO3sI
sinu cosv sinu sin v
2
ổ ử
p
ữ
ỗ
ữ
= = -
ỗ
ữ
ỗ
ữ
ỗ
ố ứ

( )
u v k2 u v k2 , k
2 2
p p
= - + p = + + p ẻ Â

w6c7O&ZR63!
-
.(.\
2
sinx cos x
3
ổ ử
p
ữ
ỗ
ữ
= -
ỗ
ữ
ỗ
ữ
ỗ
ố ứ

3\agH$^2'Q7[[(

x$
-
N6K7@HI

( )
( )
sin x k2 sinx
cos x k2 cosx
ỡ
ù
+ p =
ù
ù
ớ
ù
+ p =
ù
ù
ợ

( )
( )
( )

sin x k2 sinx
k
cos x k2 cosx
ỡ
ù
+p + p = -
ù
ù
ẻ

ớ
ù
+p + p = -
ù
ù
ợ
Â

8AP;4QR;8AB;449>PS;
(=T
u v k2
sinu sinv
u v k2
ộ
= + p
ờ
=
ờ
= p - + p
ờ
ở
"!
-
3
-
I
sinx 0 x k
sinx 1 x k2
2
sinx 1 x k2

2
ỡ
ù
ù
= ị = p
ù
ù
ù
ù
p
ù
= ị = + p
ớ
ù
ù
ù
p
ù
ù
= - ị = - + p
ù
ù
ợ
(=T
u v k2
cosu cosv
u v k2
ộ
= + p
ờ

=
ờ
= - + p
ờ
ở
"!
-
3
-
I
cosx 0 x k
2
cosx 1 x k2
cosx 1 x k2
ỡ
ù
p
ù
= ị = + p
ù
ù
ù
ù
= ị = p
ớ
ù
ù
= - ị = p + p
ù
ù

ù
ù
ợ
(=T
tanu tanv u v k
ék : u,v k
2
= = + p
p
ạ + p
"!
-
3
-
I
tanx 0 x k
tanx 1 x k
4
ỡ
ù
= = p
ù
ù
ù
ớ
p
ù
= = + p
ù
ù

ù
ợ
?#%@3@$'AAAA? B+
!"
(=T
cotu cot v u v k
ék : u,v k
= = + p
ạ p
"!
-
3
-
I
cot x 0 x k
2
cot x 1 x k
4
ỡ
ù
p
ù
= = + p
ù
ù
ù
ớ
ù
p
ù

= = + p
ù
ù
ù
ợ
U9V

>(;4
Wp2.I
( )
cos3x 4cos2x 3cosx 4 0 , x 0;14
ộ ự
- + - = * " ẻ
ờ ỳ
ở ỷ
Xp2.I
( ) ( ) ( )
2cosx 1 2sinx cosx sin2x sinx- + = - *
Yp2.I
( )
cos3x cos2x cosx 1 0+ - - = *
Zp2.I
( )
sinx cosx 1 sin2x cos2x 0+ + + + = *
[p2.I
( ) ( )
2sinx 1 cos2x sin2x 1 cosx+ + = + *
\p2.I
( )

1 1 7
4sin x
sinx 4
3
sin x
2
ổ ử
p
ữ
ỗ
ữ
+ = - *
ỗ
ữ
ỗ
ổ ử
ữ
ỗ
p
ố ứ
ữ
ỗ
ữ
-
ỗ
ữ
ỗ
ữ
ỗ
ố ứ

]p2.I
( )

4 4
7
sin x cos x cot x cot x
8 3 6
ổ ử ổ ử
p p
ữ ữ
ỗ ỗ
ữ ữ
+ = + - *
ỗ ỗ
ữ ữ
ỗ ỗ
ữ ữ
ỗ ỗ
ố ứ ố ứ
^p2.I
( )

4 4
4
sin 2x cos 2x
cos 4x
tan x tan x
4 4
+
= *

ổ ử ổ ử
p p
ữ ữ
ỗ ỗ
ữ ữ
- +
ỗ ỗ
ữ ữ
ỗ ỗ
ữ ữ
ỗ ỗ
ố ứ ố ứ
_p2.I
( )

3 x 1 3x
sin sin 1
10 2 2 10 2
ổ ử ổ ử
p p
ữ ữ
ỗ ỗ
ữ ữ
- = +
ỗ ỗ
ữ ữ
ỗ ỗ
ữ ữ
ỗ ỗ
ố ứ ố ứ

W`p2.I
( )
sin 3x sin2xsin x 1
4 4
ổ ử ổ ử
p p
ữ ữ
ỗ ỗ
ữ ữ
- = +
ỗ ỗ
ữ ữ
ỗ ỗ
ữ ữ
ỗ ỗ
ố ứ ố ứ
WW
( )

3
8cos x cos3x 1
3
ổ ử
p
ữ
ỗ
ữ
+ =
ỗ
ữ

ỗ
ữ
ỗ
ố ứ
WXp2.I
( )

3
2sin x 2sinx 1
4
ổ ử
p
ữ
ỗ
ữ
+ =
ỗ
ữ
ỗ
ữ
ỗ
ố ứ
WYp2.I
( )

3
sin x 2sinx 1
4
ổ ử
p

ữ
ỗ
ữ
- =
ỗ
ữ
ỗ
ữ
ỗ
ố ứ
WZp2.I
( )
cosx cos2x cos3x cos4x 0+ + + = *
W[p2.I
( )

2 2 2
3
sin x sin 2x sin 3x
2
+ + = *

W\p2.I
( )

2 2 2
sin x sin 2x sin 3x 2+ + = *

W]p2.I
( )

2 2 2 2
sin x sin 3x cos 2x cos 4x+ = + *
W^p2.I
( )

2 2 2 2
sin 3x cos 4x sin 5x cos 6x- = - *
B: ?EBFGBF'GBBBBHF7AA?
  !
"
W_p2.I
( )
 
2 2
5x 9x
cos3x sin7x 2 2cos
4 2 2
æ ö
p
÷
ç
÷
+ = + - *
ç
÷
ç
÷
ç
è ø

X`p2.I
( )

2 2 2
sin x cos 2x cos 3x= + *
XWp2.I
( )

2
2sin 2x sin7x 1 sinx+ - = *
XXp2.I
( )
sinx sin2x sin3x 1 cosx cos2x+ + = + + *
XYp2.I
( )

3 3 3
sin xcos3x cos xsin3x sin 4x+ = *
XZp2.I
( )

2 3
cos10x 2cos 4x 6cos3xcosx cosx 8cosxcos 3x+ + = + *
X[p2.I
( )

3 3 2
4sin x 3cos x 3sinx sin xcosx 0+ - - = *
X\p2.I
( ) ( ) ( )


2
2sinx 1 3cos4x 2sinx 4 4cos x 3+ + - + = *
X]p2.I
( )
( )

6 6 8 8
sin x cos x 2 sin x cos x+ = + *
X^p2.I
( )
( )

8 8 10 10
5
sin x cos x 2 sin x cos x cos2x
4
+ = + + *
X_p2.I
( )
( )

3 3 5 5
sin x cos x 2 sin x cos x+ = + *
Y`p2.I
( )

4 2 2 4
3cos x 4cos xsin x sin x 0- + = *
YWp2.I

( )

3 3
2 3 2
cos3xcos x sin3xsin x
8
-
- = *
YXp2.I
( )

1
cosxcos2xcos4xcos8x
16
= *
YYp2.I
( )

3
4sin3xcos2x 1 6sinx 8sin x= + - *
YZp2.I
( )

1
cosx cos2x cos3x cos4x cos5x
2
+ + + + = - *
Y[p2.I
( )


sin2x 2cosx sinx 1
0
tanx 3
+ - -
= *
+
Y\p2.I
( )

2
1 sin2x cos2x
2sinxsin2x
1 cot x
+ +
= *
+
Y]p2.I
( ) ( )
tanx cotx 2 sin2x cos2x+ = + *
Y^p2.I
( )

2
tan x tanxtan3x 2- = *
Y_p2.I
( )

2 2 2
11
tan x cot x cot 2x

3
+ + = *
Z`p2.I
( )

2 2 2
x x
sin tan x cos 0
2 4 2
æ ö
p
÷
ç
÷
- - = *
ç
÷
ç
÷
ç
è ø
ZWp2.I
( ) ( )

2
sin2x cotx tan2x 4cos x+ = *
ZXp2.I
( ) ( )

2 2

cot x tan x
16 1 cos4x
cos2x
-
= + *
ZYp2.I
( )

1
2tanx cot2x 2sin2x
2sin2x
+ = + *
?#%@3@$'AAAA?  B=
 !" 
ZZp2.I
( )
( ) ( )

3 sinx tanx
2 1 cosx 0
tanx sinx
+
- + = *
-
Z[p2.I
( ) ( )
( )
( ) ( )

2 2

2 2
1 cosx 1 cosx
1
tan xsinx 1 sinx tan x
2
4 1 sinx
- + +
- = + + *
-
Z\p2.I
( )
cos3xtan5x sin7x= *
Z]p2.I
( )

1 1
sin2x sinx 2cot x
2sinx sin2x
+ - - = *
Z^p2.I
( ) ( )

4 4
sin x cos x 1
tanx cot2x
sin2x 2
+
= + *
Z_p2.I
( )


2 2 2 2
tan x.cot 2x.cot3x tan x cot 2x cot3x= - + *
[`p2.I
( )

x
cot x sinx 1 tanxtan 4
2
æ ö
÷
ç
÷
+ + = *
ç
÷
ç
÷
ç
è ø
B4 ?EBFGBF'GBBBBHF7AA?
!
"
8Aa;4(b;49S98AP;4QR;8AB;449>PS;

cIy
-
U6Q
-
36

x,2x,3x
O[.sSR
-
S[J@'$
-

6SJ6
x
76
2x
u#3\KT2W6oSKXk6

-
'6I?.f\36
x,2x,3x
OSJ6
6
2x
R63T6K
*
U!
-

*
U#^H$
*
U!
-

*
UOSJ6
x
?
,2'H2
( )
( ) ( )
3 2 3 2
4cos x 3cosx 4 2cos x 1 3cosx 4 0 4cos x 8cos x 0* - - - + - = - =
( )
( )
( )
( )



2
cosx 0 N
4cos x cosx 2 0 x k , k
cosx 2 L
2
ộ
=
p
ờ
- = = + p ẻ
ờ
=
ờ
ở

Â


0,5 k 3,9
3 5 7
Do x 0;14 ,k 0 k 14 x ; ; ;
k
2 2 2 2 2
ỡ
ỡ ỹ
ù
- Ê Ê ằ
ù ù
p p p p p
ù
ù ù
ộ ự
ẻ ẻ Ê + p Ê ị ẻ
ớ ớ ý
ờ ỳ
ở ỷ
ù ù ù
ẻ
ù ù ù
ợ ỵ
ợ
Â
Â

,2'H2

( ) ( ) ( )
2cosx 1 2sinx cosx 2sinxcosx sinx* - + = -
( ) ( ) ( )
2cosx 1 2sinx cosx sinx 2cosx 1 0 - + - - =
( ) ( ) ( ) ( )
2cosx 1 2sinx cosx sinx 0 2cosx 1 sinx cosx 0
ộ ự
- + - = - + =
ờ ỳ
ở ỷ
( )

x k2
2cosx 1 0
cosx cos
3
k;l
3
sinx cosx 0
tanx 1
x l
4
ộ
p
ộ
p
ờ
= + p
ộ
- =

ờ
=
ờ
ờ
ờ
ẻ
ờ
ờ
ờ
+ = p
ờ
ờ
= -
ở
= - + p
ờ
ờ
ở
ở
Â

cIy
-
U6Q
-
6
3x

2x
O[_7SR

-
S[J@
'$
-

6U3Y$3$S$L'
,2'H2
( )
3 2 3 2
4cos x 3cosx 2cos x 1 cosx 1 0 2cos x cos x 2cosx 1 0* - + - - - = + - - =
?#%@3@$'AAAA? B8
Wp2.I
( )
cos3x 4cos2x 3cosx 4 0 , x 0;14
ộ ự
- + - = * " ẻ
ờ ỳ
ở ỷ
H d12e'f=g5!(/#X``X
Xp2.I
( ) ( ) ( )
2cosx 1 2sinx cosx sin2x sinx- + = - *
H d12e'f=g5!(/#X``Z
Yp2.I
( )
cos3x cos2x cosx 1 0+ - - = *
H d12e'f=g5!(/#X``\
Zp2.I
( )
sinx cosx 1 sin2x cos2x 0+ + + + = *

H d12e'f=g5!/#X``[
 !" 
( ) ( ) ( )
( )

2 2
cos x 2cosx 1 2cosx 1 0 2cosx 1 cos x 1 0Û + - + = Û + - =
( ) ( )
 
2
sinx 0 x k
2cosx 1 sin x 0 k;l
1 2
cosx x l2
2 3
é é
= = p
ê ê
ê ê
Û - + = Û Û Î
p
ê ê
= - = ± + p
ê ê
ë ë
¢

,2'H2
( ) ( )
2

sinx cosx 2sinxcosx 2cos x 0* Û + + + =
( ) ( )
 sinx cosx 2cosx sinx cosx 0Û + + + =
( ) ( )
 sinx cosx 1 2cosx 0Û + + =
( )
 
sinx cosx tanx 1
x k
4
k;l
1 2
2
cosx cosx cos
x l2
2 3
3
é
p
é é
= - = -
ê
= - + p
ê ê
ê
ê ê
Û Û Û Î
ê
p
ê ê

p
= - =
ê
= ± + p
ê ê
ê
ë ë
ë
¢

cIy
-
U6Q
-
L6
2x
6
x
'_SR
-
67T6
2x
J6
x
3Y$S$L'OySKU6Q
-
i6sR
( )
( )
2

sinx 1 2cos x 1 2sinxcosx 1 cosx* Û + - + = +
( ) ( )

2
2sinxcos x 2sinxcosx 1 cosx 2sinxcosx cosx 1 1 cosx 0Û + = + Û + - + =
( ) ( ) ( )
 
2
1
x k2
cosx
3
cosx 1 sin2x 1 0 k,l
2
sin2x 1
x l
4
é
p
é
ê
= ± + p
ê
= -
ê
ê
Û + - = Û Û Î
ê
ê
p

ê
=
= + p
ê
ê
ë
ë
¢

cIy
-
U6Q
-
6
3
x
2
p
-

7
x
4
p
-
[.67_SR
-
S6
H67J@'$
-

66
x
"T'SSJ6SKOKT@$
$
-
6!
-
@6%[?SNb3@b.vjjX
-
z
sSKk626S7
,2'H2
Whi$$
-
6I
( )
sin a b sina.cosb cosa.sinb± = ±
B;> ?EBFGBF'GBBBBHF7AA?
\p2.I
( )

1 1 7
4sin x
sinx 4
3
sin x
2
æ ö
p
÷

ç
÷
+ = - *
ç
÷
ç
æ ö
÷
ç
p
è ø
÷
ç
÷
-
ç
÷
ç
÷
ç
è ø
H d12e'f=g5!/#X``^
[p2.I
( ) ( )
sinx 1 cos2x sin2x 1 cosx+ + = + *
H d12e'f=g5!(/#X``^
!
"
( )
1 1 7 7

4 sin cosx sinxcos
sinx 3 3 4 4
sinxcos sin cosx
2 2
ổ ử
p p
ữ
ỗ
ữ
* + = -
ỗ
ữ
ỗ
ữ
ỗ
p p
ố ứ
-
( )

1 1 2
4. sinx cosx
sinx cosx 2
ộ ự
ờ ỳ
+ = - +
ờ ỳ
ờ ỳ
ở ỷ
"J6H

-
I
sinxcosx 0 sin2x 0ạ ạ

( )

sinx cosx
2 2 sinx cosx
sinxcosx
+
= - +
( ) ( )
sinx cosx 2 2sinxcosx sinx cosx 0 + + + =
( )
( )
sinx cosx 1 2sin2x 0 + + =
( )


x k
4
tanx 1
sinx cosx 0
x l k,l,m
2
8
1 2sin2x 0
sin2x
5
2

x m
8
ộ
p
ờ
= - + p
ờ
ộ
= -
ộ
ờ
+ =
ờ
p
ờ
ờ
ờ
= - + p ẻ
ờ
ờ
ờ
+ =
ờ
ờ= -
ờ
ở
ờ
p
ở
= + pờ

ờ
ở
Â

Xhi?
KI
( )
3
sin x sin 2 x cosx
2 2
7 1
sin x sin 2 x sin x sinx cosx
4 4 4
2
ỡ
ộ ự
ổ ử ổ ử
ù
p p
ù
ữ ữ
ỗ ỗ
ờ ỳ
ữ ữ
ù - = - p- - =
ỗ ỗ
ữ ữ
ù
ờ ỳ
ỗ ỗ

ữ ữ
ỗ ỗ
ù
ố ứ ố ứ
ờ ỳ
ù
ở ỷ
ớ
ộ ự
ổ ử ổ ử ổ ử
ù
p p p
ữ ữ ữ
ù
ỗ ỗ ỗ
ờ ỳ
ữ ữ ữ
- = p - + = - + = - +
ù
ỗ ỗ ỗ
ữ ữ ữ
ù
ờ ỳ
ỗ ỗ ỗ
ữ ữ ữ
ỗ ỗ ỗ
ố ứ ố ứ ố ứ
ù
ờ ỳ
ở ỷ

ù
ợ
( ) ( )
1 1 1
4. sinx cosx
sinx cosx
2
ộ ự
ờ ỳ
* + = - +
ờ ỳ
ở ỷ
p2X2;
cIye6
x x
3 6 2
p p p
+ + - =
[.sSR6jj.vjjO
-

-
7I
cot x cot x cot x cot x cot x tan x 1
3 6 3 2 3 3 3
ộ ự
ổ ử ổ ử ổ ử ổ ử ổ ử ổ ử
p p p p p p p
ữ ữ ữ ữ ữ ữ
ỗ ỗ ỗ ỗ ỗ ỗ

ờ ỳ
ữ ữ ữ ữ ữ ữ
+ - = + - + = + + =
ỗ ỗ ỗ ỗ ỗ ỗ
ữ ữ ữ ữ ữ ữ
ờ ỳ
ỗ ỗ ỗ ỗ ỗ ỗ
ữ ữ ữ ữ ữ ữ
ỗ ỗ ỗ ỗ ỗ ỗ
ố ứ ố ứ ố ứ ố ứ ố ứ ố ứ
ờ ỳ
ở ỷ

#$
-
^\L[@$I
4 4 2
1
sin x cos x 1 sin 2x
2
+ = -
R6
H$K
-
Uv7O'R3RSe
cos
cot
sin
=
Ofk6Sf3

sQ.\.OcSR
-
SNR6
-
'0SJ6H
-

,2'H2
?#%@3@$'AAAA? B;;
]p2.I
( )

4 4
7
sin x cos x cot x cot x
8 3 6
ổ ử ổ ử
p p
ữ ữ
ỗ ỗ
ữ ữ
+ = + - *
ỗ ỗ
ữ ữ
ỗ ỗ
ữ ữ
ỗ ỗ
ố ứ ố ứ
H d12e'f=g4$D

*h8/#W___
!"
"MI
sin x 0
1
3
sin x sin x 0 cos 2x 0 cos 2x 0
3 6 2 6 6
sin x 0
6
ỡ
ổ ử
ù
p
ữ
ù
ỗ
ữ
+ ạ
ù
ỗ
ữ
ỗ
ù
ổ ử ổ ử ổ ử ổ ử
ữ
ỗ
p p p p
ố ứ
ù

ữ ữ ữ ữ
ù
ỗ ỗ ỗ ỗ
ữ ữ ữ ữ
+ - ạ - ạ - ạ
ỗ ỗ ỗ ỗ
ớ
ữ ữ ữ ữ
ỗ ỗ ỗ ỗ
ổ ử
ữ ữ ữ ữ
ỗ ỗ ỗ ỗ
ù
p
ố ứ ố ứ ố ứ ố ứ
ữ
ỗ
ù
ữ
- ạ
ỗ
ù
ữ
ỗ
ù
ữ
ỗ
ố ứ
ù
ù

ợ

( ) ( )

2 2
1 7 1 1 k
1 sin 2x sin 2x 1 cos4x x , k
2 8 4 2 12 2
p p
* - = = - = = + ẻ Â

,2'H2
"MI
cos x 0
1
4
cos x cos x 0 cos2x cos 0 cos2x 0
4 4 2 2
cos x 0
4
ỡ
ổ ử
ù
p
ữ
ù
ỗ
ữ
- ạ
ù

ỗ
ữ
ỗ
ù
ổ ử ổ ử ổ ử
ữ
ỗ
p p p
ố ứ
ù
ữ ữ ữ
ù
ỗ ỗ ỗ
ữ ữ ữ
- + ạ + ạ ạ
ỗ ỗ ỗ
ớ
ữ ữ ữ
ỗ ỗ ỗ
ổ ử
ữ ữ ữ
ỗ ỗ ỗ
ù
p
ố ứ ố ứ ố ứ
ữ
ỗ
ù
ữ
+ ạ

ỗ
ù
ữ
ỗ
ù
ữ
ỗ
ố ứ
ù
ù
ợ

KI
tan x tan x tan x tan x tan x cot x 1
4 4 4 2 4 4 4
ộ ự
ổ ử ổ ử ổ ử ổ ử ổ ử ổ ử
p p p p p p p
ữ ữ ữ ữ ữ ữ
ỗ ỗ ỗ ỗ ỗ ỗ
ờ ỳ
ữ ữ ữ ữ ữ ữ
- + = - - + = - - =
ỗ ỗ ỗ ỗ ỗ ỗ
ữ ữ ữ ữ ữ ữ
ờ ỳ
ỗ ỗ ỗ ỗ ỗ ỗ
ữ ữ ữ ữ ữ ữ
ỗ ỗ ỗ ỗ ỗ ỗ
ố ứ ố ứ ố ứ ố ứ ố ứ ố ứ

ờ ỳ
ở ỷ

( )
( )
2 4 2 4 4 2
1 1
1 sin 4x cos 4x 1 1 cos 4x cos 4x 2cos x cos 4x 1 0
2 2
* - = - - = - - =
( )
( )



2
2 2
2
2
t 1 N
2t t 1 0
1
cos 4x 1 sin 4x 0 sin4x 0
t L
t cos 4x 0
2
t cos 4x 0
ỡ
ộ
ù

=
ù
ờ
ù
ỡ
ùù
ờ
- - =
ù
ù
ù ù
ờ
= = =
ớ ớ
= -
ờ
ù ù
=
ở
ù ù
ù
ợ
ù
ù
=
ù
ù
ợ
( )
( )

( )



sin2x 0 N
k
x , k
cos2x 0 L
2
ỡ
ù
=
p
ù
ù
= ẻ
ớ
ù
=
ù
ù
ợ
Â

6zOKTX
-
3RSe'16N3Y$$
-
B6
tan tanx tan tanx

1 tanx 1 tanx
4 4
tan x .tan x . . 1
4 4 1 tanx 1 tanx
1 tan tanx 1 tan tanx
4 4
p p
- +
ổ ử ổ ử
p p - +
ữ ữ
ỗ ỗ
ữ ữ
- + = = =
ỗ ỗ
ữ ữ
ỗ ỗ
ữ ữ
ỗ ỗ
p p + -
ố ứ ố ứ
+ -

cI.7O_@$$
-
6BAAO!
-
Uv
e6L[OAASy$
-

'ROKgHKSSRHRk62n7
UB'(6
3 x
10 2
p
-

3x
10 2
p
+
K'N
-
7H$u
-

-
7I
B;* ?EBFGBF'GBBBBHF7AA?
^p2.I
( )

4 4
4
sin 2x cos 2x
cos 4x
tan x tan x
4 4
+
= *

ổ ử ổ ử
p p
ữ ữ
ỗ ỗ
ữ ữ
- +
ỗ ỗ
ữ ữ
ỗ ỗ
ữ ữ
ỗ ỗ
ố ứ ố ứ
H d12e'f=gi2(7/#W__]
_p2.I
( )

3 x 1 3x
sin sin 1
10 2 2 10 2
ổ ử ổ ử
p p
ữ ữ
ỗ ỗ
ữ ữ
- = +
ỗ ỗ
ữ ữ
ỗ ỗ
ữ ữ
ỗ ỗ

ố ứ ố ứ
H d12e'f=gj2/#X``W
!
"
3x 3x 9 3x 3 x
sin sin sin sin3
10 2 10 2 10 2 10 2
ộ ự
ổ ử ổ ử ổ ử ổ ử
p p p p
ữ ữ ữ ữ
ỗ ỗ ỗ ỗ
ờ ỳ
ữ ữ ữ ữ
+ = p- + = - = -
ỗ ỗ ỗ ỗ
ữ ữ ữ ữ
ờ ỳ
ỗ ỗ ỗ ỗ
ữ ữ ữ ữ
ỗ ỗ ỗ ỗ
ố ứ ố ứ ố ứ ố ứ
ờ ỳ
ở ỷ
ySKOg
S!
-

3 x
t

10 2
p
= -
i$3N6Q
,2'H2
KI
3x 3x 9 3x 3 x
sin sin sin sin3
10 2 10 2 10 2 10 2
ộ ự
ổ ử ổ ử ổ ử ổ ử
p p p p
ữ ữ ữ ữ
ỗ ỗ ỗ ỗ
ờ ỳ
ữ ữ ữ ữ
+ = p- + = - = -
ỗ ỗ ỗ ỗ
ữ ữ ữ ữ
ờ ỳ
ỗ ỗ ỗ ỗ
ữ ữ ữ ữ
ỗ ỗ ỗ ỗ
ố ứ ố ứ ố ứ ố ứ
ờ ỳ
ở ỷ

( ) ( )

3 x 1 3 x

1 sin sin3 2
10 2 2 10 2
ổ ử ổ ử
p p
ữ ữ
ỗ ỗ
ữ ữ
- = -
ỗ ỗ
ữ ữ
ỗ ỗ
ữ ữ
ỗ ỗ
ố ứ ố ứ

"!
-

3 x
t
10 2
p
= -

( )
( ) ( )
3 2
1 1
2 sint sin3t sint 3sint 4sin t sint 1 sin t 0
2 2

= = - - =
( )

3 x 3
t k
k x k2
sint 0
10 2 5
k,l
cost 0 3 x 2
t l
l x l2
2
10 2 2 5
ộ ộ
p p
ộ
= p
ờ ờ
- = p = - p
ộ
=
ờ
ờ ờ
ờ
ờ
ẻ
ờ ờ
p
ờ

ờ
= p p p
= + p
ờ ờ
ờ
ở
- = + p = - p
ờ
ờ ờ
ở
ở ở
Â

,2'H2
KI
3
sin 3x sin 3x sin 3x sin 3x sin3 x
4 4 4 4 4
ộ ự
ổ ử ổ ử ổ ử ộ ự ổ ử
p p p p p
ữ ữ ữ ữ
ỗ ỗ ỗ ỗ
ờ ỳ
ờ ỳ
ữ ữ ữ ữ
- =- - = - p- - = - + =- +
ỗ ỗ ỗ ỗ
ữ ữ ữ ữ
ờ ỳ

ỗ ỗ ỗ ỗ
ờ ỳ
ữ ữ ữ ữ
ỗ ỗ ỗ ỗ
ố ứ ố ứ ố ứ ố ứ
ờ ỳ
ở ỷ
ở ỷ
"!
-

t x x t
4 4
p p
= + ị = -
[SK
( )
1 sin3t sin 2t .sint
2
ổ ử
p
ữ
ỗ
ữ
- = -
ỗ
ữ
ỗ
ữ
ỗ

ố ứ
3
2 2 2
sint 0 sint 0
4sin t 3sint cos2tsint 0
4sin t 3 1 2sin t 0 sin t 1
ộ ộ
= =
ờ ờ
- + =
ờ ờ
- + - = =
ờ ờ
ở ở
( )

t k
x k
sint 0
4
x m , k,l,m
cost 0
4 2
t l
x l
2
4
ộ
p
ộ

= p
ờ
= - + p
ộ
=
ờ
p p
ờ
ờ
ờ
= - + ẻ
ờ
p
ờ
ờ
= p
= + p
ờ
ờ
ở
= + p
ờ
ờ
ở
ở
Â

,2'H2
KI
( )

cos3x cos 3x cos 3 x
3
ộ ự
ổ ử
p
ữ
ỗ
ờ ỳ
ữ
= - p + =- +
ỗ
ữ
ờ ỳ
ỗ
ữ
ỗ
ố ứ
ờ ỳ
ở ỷ

?#%@3@$'AAAA? B;9
WWp2.I
( )

3
8cos x cos3x 1
3
ổ ử
p
ữ

ỗ
ữ
+ =
ỗ
ữ
ỗ
ữ
ỗ
ố ứ
H d12e'f=gk1!48;$

/#W___
W`p2.I
( )
sin 3x sin2xsin x 1
4 4
ổ ử ổ ử
p p
ữ ữ
ỗ ỗ
ữ ữ
- = +
ỗ ỗ
ữ ữ
ỗ ỗ
ữ ữ
ỗ ỗ
ố ứ ố ứ
H d12e'8gD3

1HDl$/#W___
!"
I
( ) ( )

3
1 8cos x cos 3 x 2
3 3
ộ ự
ổ ử ổ ử
p p
ữ ữ
ỗ ỗ
ờ ỳ
ữ ữ
+ = - +
ỗ ỗ
ữ ữ
ờ ỳ
ỗ ỗ
ữ ữ
ỗ ỗ
ố ứ ố ứ
ờ ỳ
ở ỷ

"!
-

t x

3
p
= +
[SKI
( )
3 3 3
2 8cos t cos3t 8cos t 4cos t 3cost = - = - +
( )
( )

3 2
12cos t 3cost 0 cos3t 4cos t 1 0 cos3t 2cos2t 1 0 - = - = + =
( )

t k
x k
2
cos3t 0
6
t l x l k;l;m
1
3
cos2t
2
2
x m
t m
3
3
ộ

p
ộ
ờ
= + p p
ờ
ờ
= + p
ộ
ờ
=
ờ
ờ
p
ờ
ờ
ờ
= + p = p ẻ
ờ
ờ
ờ
ờ
= -
ờ
ờ
p
ờ
ờ
ở
p
= + p

ờ
= - + pờ
ờ
ở
ờ
ở
Â

,2'H2
W
"!
-

t x x t
4 4
p p
= + ị = -
[SKI
( )
3 3
1 sin t 2sin t sin t sint cost
4
ổ ử
p
ữ
ỗ
ữ
= - = -
ỗ
ữ

ỗ
ữ
ỗ
ố ứ
( )
( ) ( )

3 3 2 2
sin t sint cost sin t sin t cos t sint cost = - = + - ã
( )

2 2
cost sin t sintcost cos t 0 - + - =
( )
( )
( )



cost 0 N
1
cost sin2t 1 0 t k x k , k
sin2t 2 L
2 2 4
ộ
ổ ử
=
p p
ờ
ữ

ỗ
ữ
- = = + p = + p ẻ
ỗ
ờ
ữ
ỗ
ữ
ỗ
=
ố ứ
ờ
ở
Â

cI
( )
ã
O$SniH_6
-
v.$
2 2
1 sin t cos t= +

-
72
.OQ6
-
6RST3Rv.Ru#62WQS
2I?M3

-
LH$Sf3
-
Hv'63
-
Ov.
2 2
1 sin t cos t= +
ST.sS2?
X
( ) ( )
3
3
1 1
1 2 . 2sin x 2sinx 2 sinx cosx 2sinx
4
2 2
ộ ự
ộ ự
ổ ử
p
ữ
ỗ
ờ ỳ
ờ ỳ
ữ
+ = + =
ỗ
ữ
ờ ỳ

ỗ ờ ỳ
ữ
ỗ
ố ứ
ờ ỳ
ở ỷ
ở ỷ
( ) ( ) ( )

3 2
sinx cosx 4sinx sinx cosx sinx cosx 4sinx + = + + =
( ) ( )
sinx cosx 1 2sinxcosx 4sinx + + =

2 2
3sinx 2cos xsinx 2sin xcosx cosx 0 - + + + =
( ) ( )

2 2
sinx 3 2cos x cosx 2sin x 1 0 - + + + =
( ) ( )

2 2
sinx 2sin x 1 cosx 2sin x 1 0 - + + + =
( )
( )
( )


2

2
0 2sin x 1 0 VN
2sin x 1 cosx sinx 0
cosx sinx 0
ộ
= + >
ờ
+ - =
ờ
- =
ờ
ở
B;< ?EBFGBF'GBBBBHF7AA?
WXp2.I
( )

3
2sin x 2sinx 1
4
ổ ử
p
ữ
ỗ
ữ
+ =
ỗ
ữ
ỗ
ữ
ỗ

ố ứ
H d12e'D3

H12d$/#W__^
!
"
( )
tanx 1 x k , k
4
p
= = + p ẻ Â

Y
( ) ( )
3
3
1 1
1 2 . 2sin x 2sinx 2 sinx cosx 2sinx
4
2 2
ộ ự
ộ ự
ổ ử
p
ữ
ỗ
ờ ỳ
ờ ỳ
ữ
+ = + =

ỗ
ữ
ờ ỳ
ỗ ờ ỳ
ữ
ỗ
ố ứ
ờ ỳ
ở ỷ
ở ỷ
( ) ( )

3
sinx cosx 4sinx 2 + =

( )
cosx 0 hay sinx 1= =
H$.2
-
'L.
( )
2
RL
.
( )
2

3
cos x
OSI

( ) ( )
( )
3
2
2 tanx 1 4tanx. 1 tan x + = +
p2.BUS
-
'I
( )
tanx 1 x k , k
4
p
= = + p ẻ Â

,2'H2
W
"!
-

t x x t
4 4
p p
= - ị = +
[SKI
( )
( )
3 3
1 sin t 2sin t 4 sin t sint cost = + = +
( )
( )

3 2 2
sin t sint cost sin t cos t = + +
( )

3 3 2 2 3
sin t sin t sintcos t costsin t cos t cost sintcost 1 0 = + + + + =
( )
( )


cost 0
3
t k x k k , k
sin2t 2 L
2 4 4
ộ
=
p p p
ờ
= + p = + p - + p ẻ
ờ
= -
ờ
ở
Â

XYXc;9,\SaX2
cI,K6H6B'$
-

'3
-
Q!
-
!
-
2
\e!
-

-
6v.N\7!
-
.
-
6
!
-
e6L[3Y6OW.7STz
( )
x 4x 5x+ =

( )
2x 3x 5x+ =
\.2v.
-
7uzQS2O
[%(?yN6?STK'OS3J\.
cN
,2'H2

( ) ( ) ( )
5x 3x 5x x
cosx cos4x cos2x cos3x 0 2cos cos 2cos cos 0
2 2 2 2
* + + + = + =

5x 3x x 5x x
2cos cos cos 0 4cos cosxcos 0
2 2 2 2 2
ổ ử
ữ
ỗ
ữ
+ = =
ỗ
ữ
ỗ
ữ
ỗ
ố ứ
?#%@3@$'AAAA? B;+
WZp2.I
( )
cosx cos2x cos3x cos4x 0+ + + = *

WYp2.I
( )

3
sin x 2sinx 1

4
ổ ử
p
ữ
ỗ
ữ
- =
ỗ
ữ
ỗ
ữ
ỗ
ố ứ
H d12e'f=gk1!4*h8/#W__^
 !" 
( )
 
5x k2
5x k x
cos 0
2 2 5 5
2
cosx 0 x l x l k;l;m
2 2
x
x x 2m
cos 0
m
2
2 2

é é
p p p
é
ê ê
= + p = +
ê
ê ê
=
ê
ê ê
p p
ê
ê ê
Û = Û = + p Û = + p Î
ê
ê ê
ê
ê ê
ê
ê ê
p = p + p
=
ê
= + p
ê ê
ê
ë
ê ê
ë ë
¢


cI 0(.K(\i3
-
BOW@$
\3
-
ST3sS2
,2'H2
( ) ( ) ( ) ( ) ( )
1 1 1 3
1 cos2x 1 cos4x 1 cos6x cos2x cos6x cos4x 0
2 2 2 2
* Û - + - + - = Û + + =
( )
 2cos4xcos2x cos4x 0 cos4x 2cos2x 1 0Û + = Û + =
( )
 
k
cos4x 0
4x k x
2 8 4
k,l
2
2
cos2x cos
2x l2 x l
3
3 3
é é
p p p

é
=
ê ê
= + p = +
ê
ê ê
ê
Û Û Û Î
ê ê
p
ê
p p
=
ê ê
= ± + p = ± + p
ê
ê ê
ë
ë ë
¢

,2'H2
( ) ( ) ( ) ( )
1 1 1
1 cos2x 1 cos4x 1 cos6x 2
2 2 2
* Û - + - + - =
( ) ( ) ( )

1 1

cos2x cos4x cos6x cos2x cos6x cos4x 1 0
2 2
Û - + + = Û + + + =
( )

2
2cos4xcos2x 2cos 2x 0 2cos2x cos4x cos2x 0Û + = Û + =
( )
 
x k
2
cosx 0
4cos2xcos3xcosx 0 cos2x 0 x l k,l,m
4 2
cos3x 0
x m
6 3
é
p
ê
= + p
ê
é
=
ê
ê
p p
ê
ê
Û = Û = Û = + Î

ê
ê
ê
ê
=
ê
ê
p p
ë
= +
ê
ê
ë
¢

,2'H2
B;: ?EBFGBF'GBBBBHF7AA?
W[p2.I
( )

2 2 2
3
sin x sin 2x sin 3x
2
+ + = *

H d12e' )L=#8m35!/#X```
W\p2.I
( )


2 2 2
sin x sin 2x sin 3x 2+ + = *

H d12e'f=gL=#no1

*h85!/#X``W
W]p2.I
( )

2 2 2 2
sin x sin 3x cos 2x cos 4x+ = + *
H d12e'f=gnFk1!(/#W___
  !
"
( ) ( ) ( ) ( ) ( )
1 1 1 1
1 cos2x 1 cos6x 1 cos4x 1 cos8x
2 2 2 2
* Û - + - = + + +
( )
 cos2x cos6x cos4x cos8x 2cos4xcos2x 2cos6xcos2xÛ - + = + Û - =
( )
 2cos2x cos6x cos4x 0 4cos2xcos5xcosx 0Û + = Û =
( )

      
cosx 0
m
cos2x 0 x k x l x ; k,l,m
2 4 2 10 5

cos5x 0
é
=
ê
æ ö æ ö æ ö
p p p p p
ê
÷ ÷ ÷
ç ç ç
÷ ÷ ÷
Û = Û = + p Ú = + Ú = + Î
ç ç ç
ê
÷ ÷ ÷
ç ç ç
÷ ÷ ÷
ç ç ç
è ø è ø è ø
ê
=
ê
ë
¢

,2'H2
( ) ( ) ( ) ( ) ( )
1 1 1 1
1 cos6x 1 cos8x 1 cos10x 1 cos12x
2 2 2 2
* Û - - + = - - +

 cos6x cos8x cos10x cos12x 2cos7xcosx 2cos11xcosxÛ + = + Û =
( ) ( )
 
x k
2
cosx 0
l
cosx cos7x cos11x 0 x k,l,m
cos7x cos11x
2
m
x
9
é
p
ê
= + p
ê
ê
é
=
p
ê
ê
Û - = Û Û =- Î
ê
ê
=
ê
ê

ë
ê
p
=
ê
ê
ë
¢

,2'H2
( )
U Ucos3x sin7x 1 cos 5x 1 cos9 cos3x sin7x sin5x cos9
2
æ ö
p
÷
ç
÷
* Û + = - + - - Û + = -
ç
÷
ç
÷
ç
è ø
 cos3x cos9x sin7x sin5x 0 2cos6xcos3x 2cos6xsinx 0Û + + - = Û + =
( ) ( )

x k
12 6

cos6x 0
cos6x cos3x sinx 0 x l k,l,m
cos3x cos x
4
2
m
x
8 2
é
p p
ê
= +
ê
é
=
ê
ê
p
ê
ê
æ ö
Û + = Û Û = + p Î
p
ê
÷
çê
÷
= +
ç
ê

÷
ê
ç
÷
ç
è ø
ê
p p
ê
ë
= - +
ê
ê
ë
¢

,2'H2
?#%@3@$'AAAA?  B;=
W^p2.I
( )

2 2 2 2
sin 3x cos 4x sin 5x cos 6x- = - *
H d12e'f=g5!/#X``X
W_p2.I
( )
 
2 2
5x 9x
cos3x sin7x 2 2cos

4 2 2
æ ö
p
÷
ç
÷
+ = + - *
ç
÷
ç
÷
ç
è ø
H d12e'f=ge(Ie/#X``W
X`p2.I
( )

2 2 2
sin x cos 2x cos 3x= + *
H d12e'f=gk1!48;$

/#W__^
 !" 
( ) ( ) ( )
1 cos2x 1 cos4x 1 cos6x
cos2x cos4x 1 cos6x 0
2 2 2
- + +
* Û = + Û + + + =
( )


2
2cos3xcosx 2cos 3x 0 2cos3x cosx cos3x 0Û + = Û + =
( )
 
k
x
k
6 3
cosx 0
x
l
6 3
4cos3xcos2xcosx 0 cos2x 0 x k,l,m
l
4 2
x
cos3x 0
4 2
x m
2
é
p p
ê
= +
é
ê
é
p p
=

ê
ê
ê
= +
p p
ê
ê
ê
Û = Û = Û = + Û Î
ê
ê
ê
p p
ê
ê
ê
= +
=
ê
ê
ê
p
ë
ë
= + p
ê
ê
ë
¢


cIy
-
U6Q
-
6
( ) ( ) ( )
x , 2x , 7x

-
Uv
7x x
4x
2
+
=
OKTSV
0K'
( )
sin7x sinx-
O
( )
2
2sin 2x 1-
\06OST6H@$
ec\3
-
Y'U6Q
-
i66N@SS
.cNS2

,2'H2
( ) ( )
( )
2
sin7x sinx 1 2sin 2x 0 2cos4xsin3x cos4x 0* Û - - - = Û - =
( ) ( )
 
k2
cos4x 0
x
18 3
cos4x 2sin3x 1 0 k,l
1
5 l2
sin3x
x
2
18 3
é
p p
é
=
ê
= +
ê
ê
ê
Û - = Û Û Î
ê
ê

p p
=
ê
= +
ê
ê
ë
ë
¢

,2'H2
( ) ( ) ( )
sinx sin3x sin2x 1 cos2x cosx* Û + + = + +
( ) ( )

2
2sin2xcosx sin2x 2cos x cosx sin2x 2cosx 1 cosx 2cosx 1 0Û + = + Û + - + =
( ) ( ) ( ) ( )
 2cosx 1 sin2x cosx 0 2cosx 1 2sinxcosx cosx 0Û + - = Û + - =
( ) ( ) ( )



  
x k
2
cosx 0
x l2
1
6

cosx 2sinx 1 2cosx 1 0 sinx k,l,m,n
5
2
x m2
1
6
cosx
2
2
x n2
3
é
p
ê
= + p
é
ê
ê
ê
=
é
p
ê
ê
ê
= + p
ê
ê
ê
ê

ê
Û - + = Û = Û Î
ê
ê
ê
p
ê
ê
ê
= + p
ê
ê
ê
ë
= -
ê
ê
p
ë
ê
= ± + p
ê
ë
¢

B;4 ?EBFGBF'GBBBBHF7AA?
XXp2.I
( )
sinx sin2x sin3x 1 cosx cos2x+ + = + + *
XYp2.I

( )

3 3 3
sin xcos3x cos xsin3x sin 4x+ = *
H d12e'f=g;=/#W___
XWp2.I
( )

2
2sin 2x sin7x 1 sinx+ - = *
H d12e'f=g/#5!/#X``]
  !
"
,2'H2
( )
( ) ( )
3 3 3 3 3
sin x 4cos x 3cosx cos x 3sinx 4sin x sin 4x* Û - + - =

3 3 3 3 3 3 3
4sin xcos x 3sin x3cosx 3cos xsinx 4cos xsin x sin 4xÛ - + - =
( )

2 2 3
3sinxcosx cos x sin x sin 4xÛ - =

3 3
3 3
sin2xcos2x sin 4x sin4x sin 4x
2 4

Û = Û =
( )
  
3
k
3sin4x 4sin 4x 0 sin12x 0 12x k x , k
12
p
Û - = Û = Û = p Û = Î ¢

,2'H2
( ) ( )
( )
3
cos10x 1 cos8x cosx 2cosx 4cos 3x 3cos3x* Û + + = + -
( )
 cos10x cos8x 1 cosx 2cosxcos9xÛ + + = +
 2cos9xcosx 1 cosx 2cosxcos9xÛ + = +
( )
 cosx 1 x k2 , kÛ = Û = p Î ¢

,2'H2
( )
( ) ( )
2 2 2
sinx 4sin x 3 cosx sin x 3cos x 0* Û - - - =
( ) ( )

2 2 2
sinx 4sin x 3 cosx sin x 3 1 sin x 0

é ù
Û - - - - =
ê ú
ë û
( )
( )

2
4sin x 3 sinx cosx 0Û - - =
( ) ( )
 2 1 cos2x 3 sinx cosx 0
é ù
Û - - - =
ê ú
ë û
( )

2
1 2 2
x k
cos2x cos 2x k2
3
k;l
2 3 3
sinx cosx tanx 1
x l
4
é
p
é é

p p
ê
= ± + p
ê ê
= - = = ± + p
ê
ê ê
Û Û Û Î
ê
ê ê
p
ê
= =
= + p
ê ê
ê
ë ë
ë
¢

,2'H2
( ) ( ) ( )
( )
2
2sinx 1 3cos4x 2sinx 4 4 1 sin x 3 0* Û + + - + - - =
( ) ( ) ( ) ( )
 2sinx 1 3cos4x 2sinx 4 1 2sinx 1 2sinx 0Û + + - + - + =
( ) ( )
 2sinx 1 3cos4x 2sinx 4 1 2sinx 0Û + + - + - =
?#%@3@$'AAAA?  B;8

XZp2.I
( )

2 3
cos10x 2cos 4x 6cos3xcosx cosx 8cosxcos 3x+ + = + *
X[p2.I
( )

3 3 2
4sin x 3cos x 3sinx sin xcosx 0+ - - = *
X\p2.I
( ) ( ) ( )

2
2sinx 1 3cos4x 2sinx 4 4cos x 3+ + - + = *
 !" 
( ) ( )
 3 cos4x 1 2sinx 1 0Û - + =
( )
 
k
x
4x k2
2
cos4x 1
x l2 x l2 k;l;m
1
6 6
sin2x
2

7 7
x m2 x m2
6 6
é
p
é
ê
=
ê
= p
ê
ê
é
=
ê
ê
ê
p p
ê
ê
ê
Û Û = - + p Û = - + p Î
ê
ê
ê
= -
ê
ê
ê
ê

ë
p p
ê
= + p = + p
ê
ê
ê
ë
ë
¢

,2'H2
( )
( ) ( )
6 8 6 8 6 2 6 2
sin x 2sin x cos x 2cos x 0 sin x 1 2sin x cos x 2cos x 1 0* Û - + - = Û - - - =
( )

6 6 6 6
sin xcos2x cos xcos2x 0 cos2x sin x cos x 0Û - = Û - =
( )

6 6
k
x
cos2x 0
cos2x 0
k
4 2
x , k

tanx 1
sin x cos x
4 2
x k
4
é
p p
ê
é
= +
é
=
=
p p
ê
ê
ê
Û Û Û Û = + Î
ê
ê
ê
= ± p
=
ê
ê
ê
ë
ë = ± + p
ê
ë

¢

,2'H2
( )
( ) ( )
10 8 8 10
5
2cos x cos x sin x 2sin x cos2x 0
4
* Û - - - + =
( ) ( )

8 2 8 2
5
cos x 2cos x 1 sin x 1 2cos x cos2x 0
4
Û - - - + =

8 8 8 8
5 5
cos x.cos2x sin xcos2x cos2x 0 cos2x cos x sin x 0
4 4
æ ö
÷
ç
÷
Û - + = Û - + =
ç
÷
ç

÷
ç
è ø
( )
( )


8 8
8 8
cos2x 0
2x k
k
2
x , k
5
5
4 2
cos x sin x 0
sin x cos x 1 VN
4
4
é
p
é
=
ê
= + p
ê
p p
ê

ê
Û Û Û = + Î
ê
ê
- + =
ê
= + >
ê
ê
ë
ë
¢

,2'H2
W
( )
( ) ( )
3 5 5 3
sin x 2sin x 2cos x cos x 0* Û - - + =
( ) ( )

3 2 3 2 3 3
sin x 1 2sin x cos x 2cos x 1 0 sin xcos2x cos xcos2x 0Û - - - = Û - =
B*> ?EBFGBF'GBBBBHF7AA?
X]p2.I
( )
( )

6 6 8 8
sin x cos x 2 sin x cos x+ = + *

H d12e'f=gk1!k18;$

n!/#W___
X^p2.I
( )
( )

8 8 10 10
5
sin x cos x 2 sin x cos x cos2x
4
+ = + + *
H d12e'f=g;=*h85!(X```
X_p2.I
( )
( )

3 3 5 5
sin x cos x 2 sin x cos x+ = + *
H d12e'f=gk1!8;$

5!(W__^
!
"
( )
( )

3 3
3
cos2x 0

m
cos2x sin x cos x 0 x , m
tan x 1
4 2
ộ
=
p p
ờ
- = = + ẻ
ờ
=
ờ
ở
Â

X
( )
( ) ( )
3 3 2 2 5 5
sin x cos x sin x cos x 2sin x 2cos x 0* + + - - =
( ) ( )

3 2 5 5 3 2
sin xcos x sin x cos x cos sin x 0 - - - =
( ) ( ) ( )

3 2 2 3 2 2 3 3
sin x cos x sin x cos x cos x sin x 0 cos2x sin x cos x 0 - - - = - =
( )

3 3 3
cos2x 0 cos2x 0
m
x , m
sin x cos x 0 tan x 1
4 2
ộ ộ
= =
p p
ờ ờ
= + ẻ
ờ ờ
- = =
ờ ờ
ở ở
Â

,2'H2
( )
2 2
2
1 cos2x 1 cos2x
3 sin 2x 0
2 2
ổ ử ổ ử
+ -
ữ ữ
ỗ ỗ
ữ ữ
* - + =

ỗ ỗ
ữ ữ
ỗ ỗ
ữ ữ
ỗ ỗ
ố ứ ố ứ
( ) ( ) ( )

2 2 2
3 1 2cos2x cos 2x 4 1 cos 2x 1 2cos2x cos 2x 0 + + - - + - + =
( )

2
8cos 2x 4cos2x 0 4cos2x 2cos2x 1 0 + = + =
( )

k
cos2x 0
x k
4 2 4
k,m
1
cos2x
x m
2
3
ộ
p p p
ộ
=

ờ
= + + p
ờ
ờ
ờ
ẻ
ờ
ờ
p
= -
ờ
= + p
ờ
ờ
ở
ở
Â

5
]
cosx 0hay sinx 1= =
H$
-
'L.
( )
*
#RL
( )
*

4
cos x
OSI
( )
2
2 4
2
t 4t 3 0
3 4tan x tan x 0
t tan x 0
ỡ
ù
- + =
ù
ù
* - + =
ớ
ù
=
ù
ù
ợ
( )

2
2
2
t 1
x k
tanx 1

tan x 1
4
t 3
k,m
tan x 3
tanx 3
x m
t tan x
3
ộ
ỡ
ộ
ù
p
=
ù
ờ
ộ
ờ
ộ
= + p
=
ù
=
ờ
ù ờ
ờ
ờ
=
ẻ

ớ
ờ
ờ
ờ
ờ
ở
ù
p
=
=
ờ
ờ
ờ
ù
= + p
ở
ở
ù
=
ờ
ù
ợ
ở
Â

,2'H2
cI
-
Q7.K
cos3x

1
sin3x
OR6i$
3STH`{SSRHRk626N@OKSN.\.
?#%@3@$'AAAA? B*;
Y`p2.I
( )

4 2 2 4
3cos x 4cos xsin x sin x 0- + = *
H d12e'f=gk1!k1*h8W__^W___ W
YWp2.I
( )

3 3
2 3 2
cos3xcos x sin3xsin x
8
-
- = *
 !" 
#cROsS7Hvv.cST.$ceKU6Q

-
N
1
2
Y'N2S0N

2 3 2

8
-
.\.3R.2L.
( ) ( ) ( )
2 2
2 3 2
cos3xcosx cos x sin3xsinx sin x
8
-
* Û - =
( ) ( )

2 2
1 1 2 3 2
cos4x cos2x cos x cos2x cos4x sin x
2 2 8
-
Û + - - =

2 2 2 2
2 3 2
cos4xcos x cos2xcos x cos2xsin x cos4xsin x
4
-
Û + - + =
( ) ( )

2 2 2 2
2 3 2
cos4x cos x sin x cos2x cos x sin x

4
-
Û + + - =
( )

2
2 3 2 1 2 3 2
cos4x cos 2x cos4x 1 cos4x
4 2 4
- -
Û + = Û + + =
( ) ( )

2 k
4cos2x 2 1 cos4x 2 3 2 cos4x x , k
2 16 2
p p
Û + + = - Û = - Û = ± + Î ¢

,2'H2
cI3U6Q
-
3N6
x,2x,4x,8x
H6O[.sSR
-

S[J@'$
-
6"T'

-
7g67_SR
-
@$
2 2
cos2x 2cos x 1 1 2sin x= - = -
OKH$H2k6'Q7O3sR
.gs.3
-
O
-
2g7HKH!STz
YO67%Q.S$6O0SR$S$L
sin
O
3Y'RL
( )
*

sinx
"TS2'.v.OHT'
UB'
sinx 0=
K.2
-
'7H$0H
C
-
Q7I
( )

 sinx 0 x k hay cosx 1 cos2x cos4x cos8x 1= Û = p = ± Û = = =

( )
1
1
16
* Û ± =
$
-
'
sinx 0 x k= Û = p
H$
-
'L
( )
*
C2*RL.
( )
*

16sinx 0¹
OSI
( )
16sinxcosxcos2xcos4xcos8x sinx 8sin2xcos2xcos4xcos8x sinx
sinx 0 sinx 0
ì ì
ï ï
= =
ï ï
* Û Û

í í
ï ï
¹ ¹
ï ï
î î

4sin4xcos4xcos8x sinx 2sin8xcos8x sinx sin16x sinx
sinx 0 sinx 0 sinx 0
ì ì ì
ï ï ï
= = =
ï ï ï
Û Û Û
í í í
ï ï ï
¹ ¹ ¹
ï ï ï
î î î

k2
x
k2
x
15
15
l
x
l
x
17 17

17 17
x m
ì
é
ï
p
ï
ê
é
=
ï
p
ï
ê
ê
=
ï
ê
ê
ï
ï
p p
Û Û
í
ê
ê
= +
ï
p p
ê

ê
ï
= +
ë
ï
ê
ï
ë
ï
¹ p
ï
ï
î
0
( )
 
17p 1
k 15n; l ; k,l,m,n,p
2
-
¹ ¹ Î ¢

B** ?EBFGBF'GBBBBHF7AA?
YXp2.I
( )

1
cosxcos2xcos4xcos8x
16
= *

H d12e'f=gnFk1!(/#W__^
YYp2.I
( )

3
4sin3xcos2x 1 6sinx 8sin x= + - *
!
"
,2'H2
( )
( )
3
4sin3xcos2x 1 2 3sinx 4sin x 4sin3xcos2x 1 2sin3x* = + - = +
( )
( )
( )

2
2sin3x 2cos2x 1 1 2sin3x 4cos x 3 1 - = - = o
]
( )
cosx 0 x k , k
2
p
= = + p ẻ Â
H$
-
'.
( )
o

OR
( )
o

cosx 0ạ
OSI
( )
( )
3
2sin3x 4cos x 3cosx cosx 2sin3xcos3x cosx - = =o
( )

l2
x
14 7
sin6x cosx cosx cos 6x l,k
m2
2
x
10 5
ộ
p p
ờ
= +
ổ ử
p
ờ
ữ
ỗ
ữ

= = - ẻ
ỗ
ờ
ữ
ỗ
ữ
ỗ
p p
ố ứ
ờ
= +
ờ
ở
Â

,2'H2
CM
( )
x k2 , k= p ẻ Â

( ) ( )
1
5
2
* = - ị *
H$K
-
'
( )
x k2 , k= p ẻ Â

CM
( )
x
x k2 , k sin 0
2
ạ p ẻ ị ạÂ
RL
( )
*

x
2sin 0
2
ạ
OSI
( )
x x x x x x
2sin cosx 2sin cos2x 2sin cos3x 2sin cos4x 2sin cos5x sin
2 2 2 2 2 2
* + + + + = -

3x x 5x 3x 7x 5x 9x 7x 11x 9x x
sin sin sin sin sin sin sin sin sin sin sin
2 2 2 2 2 2 2 2 2 2 2
- + - + - + - + - = -

( )

11x 11x 2m

sin 0 m x , m 11, m
2 2 11
p
= = p = ạ ẻ Â

,2'H2
cIM2.K!
-
OK|s'167!3
-
ZO
A.2S!
-
SJ6H
-
ST.USV"!
-
3
-
SN0(3K
!
-
On77R[3Y
sin cos
,
cos sin
ổ ử
ữ
ỗ
ữ

ỗ
ữ
ỗ
ữ
ỗ
ố ứ
Y''Sc?S
2K?t^\V'$
g@6S7STHT'UB'K
-

-
'7H$
7VU'SSJ6H
-
UB'KoH$R6o

-

-
'Q7OR6H$o\
?#%@3@$'AAAA? B*9
YZp2.I
( )

1
cosx cos2x cos3x cos4x cos5x
2
+ + + + = - *
Y[p2.I

( )

sin2x 2cosx sinx 1
0
tanx 3
+ - -
= *
+
H d12e'f=g5!(/#X`WW
!"
/!
-
3T6`a6SJ6H
-
a6
-
'@'$
-

SW^g\3oa6L
-
'HK@0a
6LSJ6H
-

/!
-
0SJ6H
-
k62.

C"J6H
-
I
tanx 3
cosx 0
ỡ
ù
ạ -
ù
ù
ớ
ù
ạ
ù
ù
ợ
( ) ( ) ( )
sin2x 2cosx sinx 1 0 2cosx sinx 1 sinx 1 0* + - - = + - + =
( ) ( ) ( )

1
x k2
cosx
3
sinx 1 2cosx 1 0 k,l
2
sinx 1
x l2
2
ộ

p
ộ
ờ
= + p
ờ
=
ờ
ờ
+ - = ẻ
ờ
ờ
p
ờ
= -
= - + p
ờ
ờ
ở
ở
Â

Ch0SJ6H
-
Oa
-
'L.
( )
x k2 , k
3
p

= + p ẻ Â

,2'H2
C"J6H
-
I
sinx 0ạ
( )
2 2
sin x(1 sin2x cos2x) 2 2sin xcosx 1 sin2x cos2x 2 2cosx* + + = + + =
( )

2
2cos x 2cosxsinx 2 2cosx 0 2cosx cosx sinx 2 0 + - = + - =
( )

cosx 0
x k
cosx 0
2
k,l
cos x 1
cosx sinx 2
x l2
4
4
ộ
ộ p
=
ờ

ộ
= + p
ờ
=
ờ
ờ
ờ
ổ ử
ẻ
p
ờ
ờ
ữ
ỗ
ờ
pữ
- =
+ =
ỗ
ờ
ờ
ữ
ờ
ỗ
= + p
ữ
ở
ỗ
ờ
ố ứ

ờ
ở
ở
Â

Ch0SJ6H
-
Oa
-
'.
( )
x k x l2 , k,l
2 4
p p
= + p = + p ẻ Â

,2'H2
C"J6H
-
I
( )
sinx 0
2sinxcosx 0 sin2x 0 2x k x k , k
cosx 0
2
ỡ
ù
ạ
p
ù

ạ ạ ạ p ạ ẻ
ớ
ù
ạ
ù
ợ
Â

( ) ( ) ( )
2 2
sinx cosx sin x cos x
2 sin2x cos2x 2 sin2x cos2x
cosx sinx sinxcosx
+
* + = + = +
( ) ( )

1 1
2 sin2x cos2x sin2x cos2x
sinxcosx sin2x
= + = +
( )

2
sin2x sin2x cos2x 1 sin 2x sin2xcos2x 1 0 + = + - =
( )

2
sin2xcos2x cos 2x 0 cos2x sin2x cos2x 0 - = - =
B*< ?EBFGBF'GBBBBHF7AA?

Y\p2.I
( )

2
1 sin2x cos2x
2sinxsin2x
1 cot x
+ +
= *
+
H d12e'f=g5!/#X`WW
Y]p2.I
( ) ( )
tanx cotx 2 sin2x cos2x+ = + *
H d12e'f=g4$D

*h8/#W__^
!
"
( )


cos2x 0
x k
4
2cos2xsin 2x 0 k,l
sin 2x 0
4
x l
4

8 2
ộ
ộ
p
=
ờ
= + p
ờ
ổ ử
p
ờ
ữ
ỗ
ờ
ổ ử
ữ
- = ẻ
ỗ
p
ờ
ữ
ữ
ỗ
ờ
ỗ
ữ
ỗ
p p
ữ
- =

ỗ
ố ứ
ờ
ữ
ờ
ỗ
= +
ữ
ỗ
ờ
ố ứ
ờ
ở
ở
Â

CMR.0SJ6H
-
O.K*a
-
'I
( )
x k x l , k,l
4 8 2
p p p
= + p = + ẻ Â

,2'H2
C"J6H
-

I
( )

3
cosx 0
k
cos3x 0 x , k
cos3x 4cos x 3cosx 0
6 3
ỡ
ù
ạ
p p
ù
ù
ạ ạ + ẻ
ớ
ù
= - ạ
ù
ù
ợ
Â

( ) ( )
sinx sinx sin3x
tanx tanx tan3x 2 2
cosx cosx cos3x
ổ ử
ữ

ỗ
ữ
* + = + =
ỗ
ữ
ỗ
ữ
ỗ
ố ứ
( )

2
sinx sinxcos3x cosxsin3x 2cos xcos3x + =
( )

2
sinxsin 2x 2cos xcos3x - =

2 2
2sin xcosx 2cos xcos3x - =
( )

2
sin x cosxcos3x do cosx 0 - = ạ
( ) ( )

1 1
1 cos2x cos4x cos2x
2 2
- - = +

( )

l
cos4x 1 x , l
4 2
p p
= - = + ẻ Â
Ch
-
'0SJ6H
-
I
#;IM
l
x
4 2
p p
= +

3 l3 2
cos3x cos 0
4 2 2
ổ ử
p p
ữ
ỗ
ữ
= + = ạ
ỗ
ữ

ỗ
ữ
ỗ
ố ứ

-

#*I,T6`a6SJ6H
-
a6
-
'OQ7H$Ka6
@6]SKI
l
x
4 2
p p
= +

-
'
L
.#*7'QJ6W
#9IR6
3 l3
3x k
4 2 2
p p p
= + = + p

3 6l 2 4k 2k 3l 0,5+ = + - =
$c
k,l ẻ Â


-
7a
-
'L.I
( )
l
x , l
4 2
p p
= + ẻ Â

?#%@3@$'AAAA? B*+
Y^p2.I
( )

2
tan x tanxtan3x 2+ = *
H d12e'f=gk1!48;$

/#W__\
l<
l:
l*
9l<

+l:
=l:
+l<
9l*
=l<
;;l:
Y_p2.I
( )

2 2 2
11
tan x cot x cot 2x
3
+ + = *