1M th

1.

cos2+sin2=1,1+cot2=cosec2 1+tan2= sec2

2. sin()=sin,cosec()=cosec , tan()=tan,cot()=cot

cos()=cossec()=sec

3. 3, 2 , ,2 2

,

F CF F

F( ) F( ) , F(2 ) F( )

3F( ) COF( ) , F( ) COF( )

2 2

4. A B (1) sin(AB)=sinAcosBcosAsinB sin(A+B)=sinAcosB+cosAsinB

cos(AB)=cosAcosB+sinAsinB cos(A+B)=cosAcosBsinAsinB

tan(AB)=tan A tan B

1 tan A tan B

tan(A+B)=

tanA tan B

1 tan A tanB

(2)

asinAx+bcosAx= 2 2a b sin(Ax+)= 2 2a b cos(Ax) tan= ba

(3)

sin2A=2sinAcosA=2

2 tanA

1 tan A

cos2A=cos2Asin2A=12sin2A=2cos2A1=

2

2

1 tan A

1 tan A

2

2 tanAtan 2A

1 tan A

Y

X

2M th

(4)

A 1 cosA A 1 cosAsin cos

2 2 2 2

A 1 cos Atan

2 1 cosA1 cos A sinA

sinA 1 cos A

(5) 3 3

3

2

sin 3A 3 sinA 4 sin A cos 3A 4 cos A 3cosA

3 tan A tan Atan 3A

1 3 tan A

(6) (7)

2 sin A cos B sin(A B) sin(A B)

2 cos A sin B sin(A B) sin(A B)

2 cos A cos B cos(A B) cos(A B)

2 sin A sin B cos(A B) cos(A B)

1 1sinA sinB 2 sin (A B) cos (A B)

2 2

1 1sinA sinB 2cos (A B) sin (A B)

2 2

1 1cosA cosB 2 cos (A B) cos (A B)

2 2

1 1cosA cosB 2 sin (A B) sin (A B)

2 2

5.

[0 , 2) sinx=a x=n+(1)

n nI

cosx=a x=2n nI

tanx=a x=n+ nI

6. ()

y=arcsinx

[1,1]

,

2 2

y=arctanx =(,),

2 2

y=arccosecx (,1][1,)

, 0 0,

2 2

y=arcosx [1,1] [0,]

y=arccotx = (,) (0,)

y=arcsecx

(,1][1,)

0, ,

2 2

3M th

1f (a) x f(x)=a

1. sin(arcsinx)= x; 1x1 arcsin(sinx)=x;x ,2 2

cos(arcosx) = x; 1x1 arcos(cosx)=x;x[0,]

tan(arctanx)= x; x arctan(tanx)=x;x ,2 2

2. arcsin(x)=arcsinx; 1x1 arccos(x)=arccosx;1x1

arctan(x)=arctanx;x arccot(x)=arccotx;x

arccosec(x)=arccosecx;x1,x1 arcsec(x)=arcsecx;x1,x1

3. arcsinx+arccosx=2

; 1x1 arcstanx+arccotx=

2

;x

4. arctanx+arctany=x y

arctan1 xy

; arctan x arctan y

2 2

arctanx+arctany=+x y

arctan1 xy

; arctan x arctan y

2

7.

2 2 2

2 2 2

2 2 2

a b c 2bc cosA

b a c 2ac cosB

c a b 2ab cosC

2 2 2

2 2 2

2 2 2

b c acos A

2bc

a c bcos B

2ac

a b ccos C

2ab

a bcosC c cosB

b = c cos A a cosC

c = a cos B b cosA

a b csin A sin B sinC

=2r

r A,B C

1ABC ab sinC2

1 ac sin B2

1 bc sin A2

ABC s(s a)(s b)(s c) a b cs2

bc

aCB

A

4M th

1. secx+tanx=A A0 sinx

1.2A 2.2

2

A 1

A 1

3.

2

A

A 1 4.

2

2A

A 1

2. 1 tan x 1 A cos x sin x1 tan x cos2x

A

1.1 2.2 3.4 4.6

3. 5cos sin3

sin2

1. 413

2.9

13 3.

4

9 4.

13

9

5M th

4. o o

o o

sin 30 cos30

sin10 cos10

1.1 2.1 3.2 4.2

5. x1 cot201 cot25

x [2]

6. x sin x cos x a sin x cos x b

sin4x

1. 3 31 (a b ab )2

2. 3 31 (ab a b)2

3. 3 3ab a b 4. 3 3a b ab

6M th

7. 1xfxx 1

x0 x1

02 2f(sec )

1. 2sin 2. 2cos 3. 2tan 4. 2cot

8. cos 36 cos 72sin 36 tan18 cos 36

9. osin15 ocos15 2x ax b 0 4a b

1. 1 2. 1 3. 2 4. 1+3 2

7M th

10.

44 44

n 1 n 144 44

n 1 n 1

cosn sin n

sin n cos n

11. n

1(sin1 )(sin 3 )(sin 5 ) (sin 89 )2

4n

8M th

12. a 5(sin a cos a) 2 sin a cos a 0.04

3 3125(sin a cos a) 75 sin a cos a

13. o o o o2 2 2 2log 1 tan1 log 1 tan2 log 1 tan 3 ... log 1 tan 44

9M th

14. A xcos x cos4

A (0, 4 )

15. o 2 o o

o 2 o

cos ec10 3 cos 70 sin 40A ,B

sec10 1 0 cos 50

2 o

o 2 o

cos 20 0C

sin 80 cos 10

det[A(B+C)]

10M th

16. R f : R R

0 ; x 1

f(x) x 1; x 1

x 1

A x R (f f)(x) cot75 1.A ( 3, 2) 2.A ( 4, 3) 3.A (2,3) 4.A (3,4)

17. 180o

11M th

18. tan20 4 sin20

sin 20 sin 40 sin 80

19. a,bR atanb

4 4

2 2

cos sin sin 2

a b ab(a b )

3 2

3a b

b 2a

12M th

20. 0x2

2 2A x log ( 3 cos x) 1 2 log (sin x)

B sec 3x cos2x x A B

21. o o0 45

tanA (sin )

cotB (sin )

sinC (cot )

cosD (cot )

1.A

13M th

22. 2 o o o o2 sin 60 (tan5 tan 85 ) 12 sin70

23.

. 3 1cos cos cos5 5 2

. 7 3tan tan cos ec16 8 8

1.. . 2.. . 3.. . 4.. .

14M th

24.

1. o ocos 75 (2 3)cos15 2. o o ocos10 sin 40 cos 20

3. A tan 3A cos 2A cos 4AcotA cos 2A cos 4A

4. A B sin2A cos2B 2sin(A B)cos(A B)

25.

a b 4 4cos sin 3sin 4 cos

a+b

15M th

26. , [ , 0] 2sin sin3

2cos cos3

1.6

2.3

3. 2

3

4. 4

3

5. 5

3

27. sin9x 6sin7x 17sin5x 12sin3xsin8x 5sin6x 12sin4x

x

16M th

28. 1A {0 x 2 | (cos2x cos 4x cos 6x) 1}3

A

29. arcsinx=2arccosx

1.1 2.2 3.3 4.4

30. 1x1 arccos x arcsin x2552

sin( )2552

1.2x 2.12 2x 3.2 2x 1 4.2x

17M th

31. arcsin(5x)+arcsin(x)=2

tan(arcsinx)

1. 15

2.1

3 3.

1

3 4.

1

2

32. 2 3(sin cos )2

04 arccos(tan 3 ) [0]

33. atanb

2 2 2 2

a acos arcsin sin arccos 1a b a b

sin

18M th

34. x arcsin x4

2sin( arccos(x ))

15

1. 1(0, )2

2.1 1( , )2 2

3.1 3

( , )22

4.3

( ,1)2

35.

71 1tan arc cot arc cot arctan5 3 9

5 12sin arcsin arcsin13 13

19M th

36. cot(arc cot 7 arc cot13 arc cot21 arc cot 31)

1. 114

2. 134

3. 92

4. 252

37. 3 5 8c arcsin arc cot arctan5 3 19

A 1 1arc cot arc cot c2x 3x

A

1. 14

2.1

4 3.

1

6 4.

1

6

20M th

38. A

2arccos(x) arccos(x 3) arccos( 1 x )

B

arccos(x) arcsin(x) arcsin(1 x)

P(S) S

P(AB)

39. 1 2arc sec x arcsin 2arccos17 5

cot arc sec x2

1. 139

2. 13

9 3. 13

16 4. 13

16

21M th

40. 2 1 1sec 2arctan arctan3 7

41. 1x1

1. sin(arccos x) cos(arcsin x) 2.arcsin x arcsin( x) 0

3.arccos x arccos( x)

4.arcsin x arccos x2

5. 1arctan x arctan , x 0x 2

42. 2 1sec (2 tan 2)

22M th

43. ABC D BC BAD CAD

BD 2CD

sin BsinC

1. 12

2.1 3.3

2 4.2

44. ABC A 60 BC= 6 AC=1

cos(2B)

1. 14

2.1

2 3.

3

2 4.

3

4

45. ABC D BC AB=4

AC=3 AD= 52

BC

1.3 2.4 3.5 4.6

23M th

46. ABC AB 2

3 3BC AC 2BC 2AC cotC [1]

1. 1

3 2. 1

2 3.1 4. 3

47. ABC a,b c A B C

1 1 1cosA cosB cosCa b c

1. 2 2 2a b c

2abc 2.

2(a b c)

abc 3.

2(a b c)

2abc 4.

2 2 2a b c

abc

24M th

48. ABC

o oABC 30 BAC 135 AD AE BAC 3

ECBC

1. 13

2. 3 3. 1

2 4. 2

49. ABC 3sin A5

5cos B13

cosC

1. 1665

2. 1665

3. 4865

4. 3365

A

B CD E

25M th

50. ABC A,B C a,b c

2 2 2a b 31c 3 tanC cotA cotB

51. ABC A,BC a,bc

(sinAsinB+sinC)(sinA+sinB+sinC)=3sinAsinC

2 23 cos e c B 3 sec B

26M th

52. ABC a,b c A B

C C o60 b=5 ac=2

ABC

1.25 2.29 3.37 4.45

53. ABC

a,b c A B C

1 1 3a c b c a b c

sinC

1. 22

2. 12

3. 32

4.1

27M th

54. a b A B ABC

2b=3a B 2A cosA

55. ABC A,B C a,b c

2 2 2a b 2555c cotCcotA cotB

28M th

1. sin 5 sin10 sin15 ... sin 355

2. cos12 cos 24 cos 36 ... cos168

3. arctan sin10 sin 20 sin 30 ... sin170

4.

44 44

k 1 k 1

44 44

k 1 k 1

cos k sin k

sin k cos k

***

5. tan18 (sin 36 sin 72 )

29M th

6. 1sin 20 sin 40 sin 80 cot102

7. sin18 sin 36 sin 54 sin 72cot9 1

8. 2 2 2

1 1 1

cos 10 sin 20 sin 40

***

9. sin9x 6sin7x 17sin5x 12sin3xsin8x 5sin6x 12sin4x

x

30M th

10. 3(1 sin )(1 cos )2

(1 sin )(1 cos )

11. n

1sin1 sin 3 sin 5 ... sin 892

4n

12. tan70 a tan10 b tan20 c tan 40 a+b+c

13. A,BR sinA+sinB=1 cosA+cosB= 12

sin2A+sin2B

31M th

14. 6 6sin cos

15. xR sin x cos x tan x cos ecx sexc cot x 8 sin2x

16. tana tanb 2x 17x 19 0

2 2sin (a b) 17 sin(a b)cos(a b) 19 cos (a b)

17. ABC oACB 120 ABC

AC AC D E BD=BE

BCE BE G CG : BD

32M th

18. ABC o A B 90 BC+CA=2(AB) cosC

19. ABC A B

cos2A+3cos2B=2 cosA 2 cos B 0 cosC

20. ABC A,B C a,b c

2 2 2a b 2555c cotCcotA cotB

33M th

1. ABC sin A : sin B : sinC 6 : 5 : 4

cosA : cosB : cosC [2:9:12]

2. O(0,0) A(1,0) B C 1

O C AB C

X X D CD 0.2 AB

[0.4]

3. ABC

(sinA+sinB+sinC)(sinA+sinBsinC)=3sinAsinB

tanC [ 3 ]

34M th

4. x,y z

2 2

2 2

2 2

x xy y 9

y yz z 16

z xz x 25

xy+yz+zx [8 3 ]

5. T(x) cos x cos 3x cos 5x ... cos 2555x x [0, ]

T(x)=0 [1277.5]