alp solutions trigonometric ratio & identities maths hindi

RESONANCE f=kd ks.kferh; vuqikr loZlfed k,¡ ,oa lehd j.k - 1

MATHEMATICS SOLUTIONS OF"ADVANCED LEVEL PROBLEMS"

Target : JEE (IITs)

TOPIC : f=kd ks.kferh; vuqikr] loZlfed k,¡ ,oa lehd j.k (TRIGONOMETRY RATIOS, IDENTITIES & EQUATIONS)

Hkkx - I

1.4

3 < < , vr%

2sin

1cot2 = cot2eccos 2

= cot21cot2

= 2)cot1( 4

3 < <

2. 2 cos x + sin x = 1 ..... (1)4 cos2 x = (1 � sinx)2

4 � 4 sin2 x = 1 + sin2 x � 2 sin x

5 sin2 x � 2 sin x � 3 = 0

(sin x � 1) (5 sin x + 3) = 0

sinx = 1, sin x = � 53

cos x = 2

xsin1 lehd j.k (1) ls

t c sin x = 1

vr% 7cos x + 6 sin x = 7

2xsin1

+ 6 sin x = 7

211

+ 6 × 1 = 6 Ans.

vkSj t c sin x = � 53

rc 7cos x + 6 sin x = 7 536

253

1

= 5

1828 = 2

3. 0° < x < 90° & cos x = 10

3

log10

sin x + log10

cos x + log10

tan x= log

10 (sin x cos x tan x)

= log10

(1 � cos2 x) = log10

(1 � 9/10)

= log10

101

= � 1

4. cot + tan = m ..... (1)sec � cos = n ..... (2)

n =

cossin2

..... (3)

m = cossin

1..... (4)

(3) / (4)


mn

= sin3

sin =

113

mn

lehd j.k (1) rFkk ABC ls

3/1

2/13/23/2

n

)nm( + 2/13/23/2

3/1

)nm(

n

= m

m2/3 � n2/3 + n2/3 = m.n1/3 (m2/3 � n2/3)1/2

1 = (mn)1/3 (m2/3 � n2/3)1/2

1 = m2/3 n2/3 (m2/3 � n2/3) m(mn2)1/3 � n(nm2)1/3 = 1

5.BsinAsin

= 23

, BcosAcos

= 25

, 0 < A, B </2

tanA = BcosBsin

5

3

tan A = 5

3 tan B ..... (1)

415

BcosBsinAcosAsin

415

Asec.Btan

Bsec.Atan2

2

lehd j.k (1) ls

5

3

415

Atan1

)Btan1(2

2

4 + 4 tan2 B = 5 + 5 tan2 A

�1 + 4 tan2 B = 5 × 53

tan2 B

tan B = ±1

tan B = +1 ( 0 < B < 2

vc tanA + tanB = 5

3 + 1 =

5

53

6. f() = sin2 + sin2

32

+ sin2

34

= 21

38

2cos134

2cos1)2cos1(

=

32

cos))22cos(22(cos)111(21

= 21

[3 � (cos 2 � cos 2)] = 23

f() = 23

(vpj) f

15 = 23


7. cos19

+ cos193

+ cos195

+...... + cos19

17

;gk¡ A = 19

, D = 192

, n = 9

cos A + cos (A + D) + cos(A + 2D) + ...... + cos (A + (n � 1)D) =

2D

sin

2nD

sin

. cos

2D)1n(A2

=

19sin

199sin

× cos

21917

19

= 199

cos

19sin

199

sin

= 21

. 21

19sin

1918

sin

. 21

19sin

19sin

8. sin sin � cos cos + 1 = 0 cos ( + ) = 1

+ = 2n

1 + cot tan =

cossinsincoscossin

=

cossin)sin(

= 0

9. (sin2 + cos2 + sin cos) � |eccos|cos

� 2 = � 1

;k 1 + sin cos � |sin | cos � 1 = 0

;k cos (sin � |sin |) = 0

2,0 ;k

,

2

ysfd u 4

(D;ksafd =4

izkUr esa ugha gS)

,

2

10. )2/cos(.)2/sin(2)2/cos(.)2/cos(2.2

= (1 + cot )2

;k

sin)cos1(2

= cosec2 + 2 cot

;k 2 + 2 cos = cosec + 2 cos

;k sin = 1/2 = n + (�1)n 6


11. 12 cos2 � 6 + 1 + cos + 2 � 2 cos2 = 0;k 10 cos2

+ cos � 3 = 0

;k (5 cos + 3) (2 cos � 1) = 0

cos = � 53

, 21

q = 3

, cos1

53

, � 3

12. cos 6x (1 + tan2x) = 1 � tan 2 x

;k cos 6x = xtan1

xtan12

2

;k cos 6x = cos 2x ;k 6x = 2n ± 2x x = 2

n,

4n

x = 0 , 4

, 4

3, ,

45

, 4

7, 2 ( x =

2

, 2

3ij tanx anx vifjHkkf"kr gks t krk gS)

13. 3cos x = |sin x|

vr% gykssa d h la[;k = 8

14. cos x = 1 � 3x2

2

2� O3

2� 3

2

5

2

3

vr% gyks d h la[;k = 5

15. x2 + y2 = 4 ............(i)tan4x + cot4x + 1 = 3 sin2y ............. (ii)

tan4x + cot4x 2vkSj sin2y 1

lehd j.k (ii) larq"V gksxh ;fn vkSj d soy ;fn tan x = cot x ,oa sin2 y = 1 gksA

tan x = ± 1 x = ± 4

vkSj sin2y = 1 y = ± 2

(lehd j.k (i) ls iznf'kZr gksus okys 'ks"k fcUnq oÙ̀k d s ckgj gksxsa) vr% oÙ̀k d s vUnj fLFkr fcUnq gksxsa&

2,

4 ,

2,

4 ,

2,

4 ,

2,

4

oÙ̀k d s vUnj fLFkr fcUnqvksa d h la[;k 4 gSA


16. cot = � 3 tan = � 3

1

cosec = � 2 sin = � 21

, IV prqFkk±'k esa gS

= 2n � 6

17.23

� 23

cos 2A + 1 � cos 2B = 1

;k23

= 23

cos 2A + cos 2B

;k 2 cos 2B + 3 cos 2A = 3

rFkk 3 sin 2A = 2 sin 2B

tan 2B = B2cosB2sin

= )A2cos1(

23

A2sin23

= Asin2

AcosAsin22 = cot AA

;k tan A tan 2B = 1 (A + 2B) = 2

18.xsin1

1xsin1

1

=

x2cos1x2cos1

xsin1xsin1

=

x2cos1x2cos1

xsin1xsin1

=

)xsin1(

xsin2

2

sin x = 21

( sinx 1)

19. tan 4

p = cot

4q

tan 4

p = tan

4

q

2 4

p= n +

2

� 4

q

4

)qp( = n +

2

p + q = 2 + 4n

20. sin x = cos y ......(1)

rFkk 6 sin y = tan z ......(2)

rFkk 2 sin z = 3 cos x ......(3)

lehd j.k (3) ls

sin2 z = 4

xcos3 2

lehd j.k (1) ls sin2x = cos2y


;k sin2y = 6

ztan2

= 61

4xcos3

1.4

xcos32

2

= )xcos34(2

xcos2

2

=

)ycos334(2

ycos12

2

=

)ycos31(2

ysin2

2

sin2y (2 + 6 cos2y � 1) = 0

sin2y = 0 ;k cos2y = � 1/6 ¼;g laHko ugha gS½ y = n sin2x = 1 rFkk sin2y = 0 ,oa sin2z = 0

21. xsin + cos x = 0

xsin = � cos x

sin x = 1 � sin2x;k sin2x + sin x � 1 = 0

;k sinx = 2

51

22. 5 � 12 tan = 11 sec 25 + 144 tan2 � 120 tan = 121 + 121 tan2

23 tan2 � 120 tan � 96 = 0

tan + tan = 23

120

tan tan = � 2396

tan ( +) =

2396

1

23120

=

119120

sin ( + ) = 169120

23. 2(sec2 � cosec2) + (cosec2 + sec2) (cosec2 � sec2) = 4

15

(cosec2 � sec2) [cosec2 + sec2 � 2] = 4

15

4(cot2 � tan2 ) (cot2 + tan2 ) = 15 4(cot4 � tan4 ) = 15 4(1 � tan8 ) = 15 tan4 4 tan8 + 15 tan4 � 4 = 0

4 tan8 + 16 tan4 � tan4 � 4 = 0

(4 tan4 � 1) (tan4 + 4) = 0

tan4 = 41

, tan4 = � 4 (vlEHko)

tan2 = ± 21

tan2 = + 21

tan = ± 2

1


24. 3 sin = sin (2 + ), (fn;k gqvk gS)tan ( + ) � 2 tan

= tan ( + ) � tan � tan

= )cos()(sin

�

cossin

� tan

= )cos(cos)sin(

� tan =

cossin

cos)cos(sin

=

cos)cos()cos(sinsin

=

cos)cos(2]sin)2[sin(sin2

=

cos)cos(2]sin3)2[sin(

= 0

25. 4 sin4x + (1 � sin2x)2 = 15 sin4x � 2 sin2x = 0sin2x (5 sin2x � 2) = 0

sin2x = 0 ; sin2x = 52

x = n ; n ;k cos 2x = 1 � 2 sin2x = 1 � 54

cos 2x = 51

= cos

2x = 2n ±

x = n ± 21

cos�1

5

1 ; n

26. 2 sin 2x cos x + sin 2x = 2 cos 2x cos x + cos 2xsin 2x (1 + 2 cos x) = cos 2x (2 cos x + 1)

(2 cos x + 1) (sin 2x � cos 2x) = 0

cos x = � 21

;k tan 2x = 1

x = 2n ± 32

; n ;k x = 2

n +

8

; n

Hkkx - II1. (i) L.H.S. = sec4 A (1 + sin2 A) (1 � sin2 A) � 2 tan2 A

= sec2 A + sec2 A sin2 A � 2 tan2 A= 1 + tan2 A + tan2 A � 2 tan2 A= 1 = R.H.S.

(ii) LHS = )sec1()sin1)(sin1(

)sec1()sin1()1(seccot2

=

2

22

cos

)1(seccot . )sec1()sin1(

= sec2 . )sec1()sin1(

= R.H.S.

2. fn;k x;k O;atd

)xcos�1(4xcos�)xsin�1(4xsin 2424 ds cjkcj gSA

= 22 )xsin�2( � 22 )xcos�2( = (2 � sin2x) � (2 � cos2x)

= cos2 x � sin2x = cos 2x


3. nh xbZ nks lehdj.ksin a � 8 sin d = 4 sin c � 7sin b,

cos a � 8 cos d = 4 cos c � 7 cos b.

mijksDr nksuksa lehdj.kksa dks oxZ dj tksMus ij1 + 64 � 16(cos a cos d + sin a sin d) = 16 + 49 � 56 (cos b cos c + sin b sin c), rFkk ;ksx lw=k yxkus ij vHkh"Vifj.kke izkIr gksxkA

4. ge tkurs gS fd cos3x = 4 cos3 x � 3 cos x, vr% 4 cos2 x � 3 = xcosx3cos

x (2k + 1). 2

, k Z, bl çdkj

(4cos2 9º � 3) (4cos2 27º � 3) = º9cosº27cos

. º27cosº81cos

= º9cosº81cos

= º9cosº9sin

= tan 9º

5. 1 + tan kº = 1 + ºkcosºksin

= ºkcos

ºksinºkcos

= ºkcos

)ºk45sin(2 =

ºkcos)ºk�45cos(2

(1 + tan kº) = ºkcos

)ºkº45cos(2

(1 + tan (45 � k)) = )ºk45cos(

ºkcos2

bl çdkj

(1 + tan kº) (1 + tan (45 � kº)) = ºkcos

)ºk�45cos(2.

)ºk�45cos()ºkcos(2

= 2

vr% izkIr gksrk gS(1 + tan 1º) (1 + tan 2º) ......(1 + tan 45º)

= (1 + tan 1º) (1 + tan 44º) (1 + tan 2º)(1 + tan43º) ......(1 + tan 22º) (1 + tan 23º) (1 + tan 45º)

= 223

n = 23

6. cos ( + ) = 54

, sin ( � ) = 135

,

2/0

4/,0

tan 2 = tan {( + ) + ( � )}

= )tan()tan(1)tan()tan(

;gk¡ tan ( + ) = 3/4, tan ( � ) = 5/12

tan 2 =

125

43

1

125

43

= 1548

56

tan 2 = 3356

7. L.H.S. = 2 sin

2.2BA

cos

2.2BA

+ cos

2C

2

= 2 sin

4

C cos

4

BA + 1 � 2 sin2

4

C

4

= 2 sin

4C

4C

4sin

4BA

cos + 1


= 1 + 2 sin

4C

4C

4cos

4BA

cos

= 1 + 2 sin

4C

2.4ABC

sin2.4

BCAsin2

= 1 + 2 sin

4

C

8

)A(2sin

8

)B(2sin2

= 1 + 4 sin

4A

sin

4B

sin

4C

= R.H.S.

8. fn;k gqvk gS (a + b)

bcos

asin 44

= 1

sin4 + cos4 + ab

sin4 + ba

cos4 = (sin2 + cos2 )2

2

2

2

2 cosba

sinab

� 2 sin2 cos2 = 0

2

22 cosba

sinab

= 0

22 cos

ba

sinab

b

cosa

sin 22

2

4

2

4

b

cos

a

sin

= (ekuk)

vc nh xbZ 'krZ d s vuqlkj

a

2

4

a

sin + b ba

1

b

cos2

4

a + b = ba

1

= 2)ba(

1

L.H.S. = 3

8

3

8

b

cos

a

sin

= a

2

2

4

a

sin

+ b

2

2

4

b

cos

= (a + b) 2

= (a + b)

2

2)ba(

1

= 3)ba(

1

= R.H.S.


9. ekuk fd P vHkh"V xq.kuQy dks n'kkZrk gS rFkk ekukQ = sin a sin 2a sin 3a.......sin 999a.

rks2999 PQ = (2 sin a cos a)(2 sin 2a cos 2a).........(2 sin 999a cos 999a)= sin 2a sin 4a .....sin 1998a= (sin 2a sin 4a ........sin 998a) [�sin(2 � 1000a)]. [�sin(2 � 1002a)]........[�sin(2 � 1998a)]

= sin 2a sin 4a ........sin 998a sin 999a sin 997a .......sin a = Q.

Q 0 vr% vHkh"V xq.kuQy P = 9992

1.

10. L.H.S. =

89

1k)º1ksin(ºksin

1=

º1sin1

89

1k

])º1kcot(º�k[cot

= º1sin

1 .cot1º = º1sin

º1cos2 fl) gqvkA

11. gesa fl) djuk gS fd2 sin 2º + 4 sin 4º +.........+ 178 sin 178º = 90 cot 1º

tks fd fuEukuqlkj gksxk2 sin 2º. sin1º + 2(2sin 4º.sin1º) +.........+ 89 (2sin 178º . sin1º) = 90 cos 1º

ge tkurs gS fd2 sin 2kº sin1º = cos(2k � 1)º � cos(2k + 1)º

vr%2 sin 2º . sin1º + 2 (2sin4º.sin1º) + 3 (2 sin 6º sin 1º) + 4(2 sin 8º sin 1º) +......+ 89 (2sin178º. sin1º)

= (cos1º � cos3º) + 2(cos3º � cos 5º) +........+ 89(cos 177º � cos179º)

= cos 1º + cos 3º + cos 5º + cos 177º + 89 cos 1º

= cos 1º + 89 cos 1º + (cos 3º + cos 5º + ...... + cos 177º)

= 90 cos 1º + 0

= 90 cos 1º

12. 4 sin 27° = 2/12/1 )53()55(

16 sin2 27 = 8(1 � cos 54°)

= 8

4

52101

= 2

52104

= 521028

= (5 + 5 ) + (3 � 5 ) � 2 )53()55(

= 2

5355

4 sin 27° =

5355

13.

2,0

iznf'kZr d juk gS & cos (sin ) > sin (cos ) sin (/2 � sin) > sin (cos )

vc gesa fl) d juk gS fd 2

� sin > cos [0, /2]

ekuk f() = /2 � sin � cos ,d Qyu gSA


f() = /2 � 2

cos

2

1sin

2

1

f() = /2 � 2 sin (/4 + )

0 /2 /4 ( + /4) /4

2

1 sin

4 1

f() = /2 � 2 sin (/4 + ) > 0 [0, /2]

/2 � sin � cos > 0 /2 � sin > cos sin (/2 � sin) > sin (cos )

cos (sin ) > sin (cos ) [0, /2]

14. ekuk x = tan A, y = tan B, z = tan C

tan (A + B + C) = AtanCtanCtanBtanBtanAtan1CtanBtanAtanCtanBtanAtan

xy + yz + zx = 1 1 � tan A tan B � tan B tan C � tan C tan A = 0

tan (A + B + C) fo|eku ugha gSA

A + B + C = (2n + 1) 2

, n I

vr% 2A + 2B + 2C = (2n + 1) tan (2A + 2B + 2C) = 0 tan 2A + tan 2B + tan 2C = tan 2A tan 2B tan 2C

Atan1

Atan22

+

Btan1

Btan22

+

Ctan1

Ctan22

=

)Ctan1)(Btan1()Atan1(

CtanBtanAtan8222

2x1

x

+ 2y1

y

+ 2z1

z

=

)z1()y1()x1(

xyz4222

15.(a) sin2 3a � sin2 a = (sin 3a + sin a)(sin 3a � sin a)

= (2sin 2a cos a)(2sin a cos 2a) = (2sin 2a cos 2a)(2sin a cos a) = sin 4a sin 2a = sin 2a sin 3a, 4a = � 3a

blfy, sin 3A = sin 4a(b) cosec a = cosec 2a + cosec 4a

sin 2a sin 4a = sin a(sin 2a + sin 4a) R.H.S. = sin a(sin 2a + sin 4a)

= sin a 2 sin 3a cos a = sin 2a sin 3a = sin 2a sin 4a = L.H.S.

(c) cos a � cos 2a + cos 3a = � (cos 6a + cos 2a + cos 4a) = �

2a2

sin

2)a2(3

sin2

a6a2cos

( 7a =

= � asin

a3sina4cos =

asin2a3cosa3sin2

= 21

(d) 3a + 4a = , ;g n'kkZrk gS fd sin 3a = sin 4a.

sin a 0 tSlk fd a = 7

sin a (3 � 4 sin2 a) = 2 sin 2a cos 2a = 4 sin a cosa cos 2a, 3 � 4 (1 � cos2 a) = 4 cos a (2 cos2 a � 1). ;g n'kkZrk gS fd


8 cos3 a � 4 cos2 a � 4 cos a + 1 = 0 .......(i)lehdj.k (i) ls ge dg ldrs gS fd cos a lehdj.k 8x3 � 4x2 � 4x + 1 = 0 dk ,d ewy gS

(e), (f), (g) rFkk (h) pw¡fd 3a + 4a = , ;g n'kkZrk gS fd tan 3a + tan 4a = 0.

a2tanatan�1a2tanatan

+ a2tan�1

a2tan22 = 0

;k tan a + 3 tan 2a � 3 tan a tan2 2a � tan3 2a = 0

ekukfd tan a = x. rc tan 2a = atan�1

atan22

= 2x�1

x2 bl çdkj

x + 2x�1

x6 � 22

3

)x�1(

x12� 32

3

)x�1(

x8 = 0

;k (1 � x2)3 + 6(1 � x2)2 � 12x2(1 � x2) � 8x2 = 0

ck;sa i{k dk izlkj djus ij mijksDr lehdj.k feyrk gSAx6 � 21x4 + 35x2 � 7 = 0 ........(i)

bl çdkj tan a mijksDr lehdj.k dk ,d ewy gS tSlk fd 6a + 8a = 2 rFkk 9a + 12a = 3 rFkk blhizdkjtan [3(2a)] + tan[4(2a)] = 0 rFkk tan [3(3a)] + tan [4(3a)] = 0 gksxkA QyLo:i tan 2a rFkk tan 3a Hkh lehdj.k (i)ds ewy gksxsaAlehdj.k (i) esa x2 = t j[kus ij gesa izkIr gksrk gS t3 � 21t2 + 35t � 7 = 0 ........(ii)

blfy, tan2 ka, k = 1,2,3 ij f=k?kkrh; lehdj.k (ii) ds fHké&fHké ewy gksxsaAtan2 a + tan2 2a + tan2 3a = 21

tan2 a tan2 2a + tan2 2a tan2 3a + tan2 a tan23a = 35

tan2 a . tan2 2a . tan2 3a = 7 tan a tan 2a tan 3a = 7

cot2a + cot22a + cot23a = a3tana2tanatan

atana3tana3tana2tana2tanatan222

222222

= 7

35 = 5

16. ;fn A + B + C = gks] rks fl) d juk gS

(i) tan²2A

+ tan²2B

+ tan²2C

1

222

2A

tan2C

tan2C

tan2B

tan2B

tan2A

tan

0

2

2C

tan2B

tan2A

tan 222 � 2

2A

tan2C

tan2C

tan2B

tan2B

tan2A

tan 0

tan 2A

tan 2B

+ tan 2B

tan an 2C

+ tan 2C

tan 2A

= 1

2

2

Ctan

2

Btan

2

Atan 222

� 2 0

tan2 2A

+ tan2 2B

+ tanan2 2C

1

(ii) sin 2A

. sin 2B

. sin 2C

81

vc sin 2A

sin 2B

sin 2C

= 21

2B

sin2A

sin2 cos

2BA

2BA

cos2

BA2

sin2C

sin


= 21

2BA

cos2

BAcos

2BA

cos

sin2A

. sin2B

. sin 2C

= 21

2BA

cos2

BAcos

2BA

cos 2 ..... (1)

= 41

[cos A + cos B � 1 � cos (A + B)]

sin2A

. sin2B

. sin 2C=

41

[cos A + cos B + cos C � 1] ..... (2)

lehd j.k (1) ls

2 sin2A

sin 2B

sin 2C

= cos

2

BA cos

2

BA � cos2

2

BA

2 sin 2A

sin 2B

sin 2C

+ cos2

2BA

� cos

2BA

cos

2BA

+ 41

cos2

2BA

� 41

cos2

2BA

= 0

2 sin 2A

sin 2B

sin 2C

+

22

2

2BA

cos41

2BA

cos21

2BA

cos

= 0

41

cos2

2BA

� 2 sin 2A

sin 2B

sin 2C

= 02

BAcos

21

2BA

cos2

2 sin 2A

sin 2B

sin 2C

41

cos2

2BA

2 sin 2A

sin 2B

sin 2C

41

( 0 < cos2

2

BA 1)

sin 2A

sin 2B

sin 2C

81

(iii) cos A + cos B + cos C 23

lehd j.k (2) ls

�1 + cos A + cos B + cos C = 4 sin 2A

sin 2B

sin 2C

81

× 4

cos A + cos B + cos C 21+ 1

cos A + cos B + cos C 23

17. tan 2 = tan

2

2 = n +

2 2 �

2 � n = 0

22 � n � 2 = 0 = 4

16nn 22

n


18. fn;k gS,5 sin x cos y = 1 vkSj 4 tan x = tan y

5 sin x cos y = 1 vkSj 4 sin x cos y = sin y cos x

sin x cos y = 51

vkSj cos x sin y = 54

sin x cos y + cos x sin y = 1

vkSj sin x cos y � cos x sin y = � 53

sin (x + y) = sin 2

vkSj sin (x � y) = sin

53

�sin 1�

x + y = n+ (�1)n

2

n Z

vkSj x � y = m+ (�1)m sin�1

53

� ; m Z

x = (m + n)2

+ (�1)n

4

+ (�1)m

21

sin�1

53

� ; m, n Z

vkSj y =(n �m)2

+ (�1)n

4

+ (�1)m+1

21

sin�1

53

� ; m, n Z

19.

2x

cos2x

sin12

2x

cos2x

sin12x

cos2x

sin

= 3

xcos

;k sin 2x

� 2x

cos = 32

cos x

;k 1 � sin x = 94

cos2x

;k94

sin2x � sin x + 1 � 94

= 0

;k 4 sin2x � 9 sin x + 5 = 0

;k (4 sin x � 5) (sin x � 1) = 0

;k sin x = 1 x = 2n + 2

20. fn;k gS

x + y = 32

vkSj cos x + cos y = 23

vr% cos x + cos y = 23

2 cos

2yx

cos

2y�x

= 23

cos

2y�x

= 23

[x + y = 32

d s ç;ksx ls ]

Li"Vr% cos

2y�x

= 23

lEHko ugha gSA

vr% fn;s x;s lehd j.k fud k; d k gy ekSt wn ugha gSA


21. ekuk 4sin x = u vkSj 31/cos y = v. vr% vc nh x;h lehd j.k fuEu : i esa gksxh&u + v= 11 ..... (i)5u2 � 2v= 2 ..... (ii)

(i) o (ii) ls vd ks foyqIr d jus ij5u2 � 22 + 2u = 2

5u2 � 22 + 2u = 2 (5u + 12) (u � 2) = 0

u � 2 = 0 [x d s leLr ekuksa d s fy;s u = 4sinx > 0 , 5u + 12 > 0] u = 2 4sinx = 2 22 sinx = 21 2 sinx = 1

sinx = 21

sinx = 21

sinx = sin 6

x = n+ (�1)n

6

; n Z

u d k eku (i) esa j[kus ij v= 9

31/cosy = 32 ycos1

= 2

cos y = 21

cos y = cos 3

y = 2m ± 3

; n Z

vr% x = n+ (�1)n

6

vkSj y = 2 m ± 3

, t gk¡ m, n Z

22. cos + sin = cos 2 + sin 2;k cos � cos 2 = sin 2 � sin

;k 2 sin 23

sin 2

= 2 sin 2

cos 23

sin 2

= 0 ;k sin 23

= cos 23

= 2n ;k = 3n2

+ 6

23. 8 sin2 x cos x = 3 sin x + cos x.

4(2 sin x cos x) sin x = 3 sin x + cos x 2(2 sin 2x sin x) = 3 sin x + cos x.

2 cos x � 2 cos 3x = 3 sin x + cos x. cos x � 3 sin x = 2 cos 3x.

cos 3x = cos

3x 3x = 2n ±

3x , n

(i) /kukRed fpUg ysus ij x = n + 6

; n

(ii) _ .kkRed fpUg ysus ij x = 2

n �

12

; n

24. sin3x cos 3x + cos2x sin 3x + 83

= 0

sin3x (3 cos3x � 3 cos x) + cos3x (3 sin x � 4 sin3x) + 83

= 0

3 sin x cos x(cos2x � sin2x) + 83

= 0 8(sin x cos x) cos 2x + 1 = 0


2 sin 4x = � 1 sin 4x = � 21

x = 4

n + (�1)n+1

6

; n

25. 3 sin x = 2(cos x + cos2x)

nksuksa i{kksa d k oxZ d jus ij3 sin2x = 4(cos2x + cos4x + 2 cos3x)

3(1 � cos2x) = 4 cos2x + 4 cos4x + 8 cos3x 4 cos4x + 8 cos3x + 7 cos2x � 3 = 0

(cos x + 1) (2 cos x � 1) (2 cos2x + 3 cos x + 3) = 0 cos x = � 1 x = (2n + 1) : n

;k cos x = 21

x = 2n ± 3

; n

26. sin4x + cos4x � 2sin2x + 43

sin22x = 0

(sin2x + cos2x)2 � 2 sin2x cos2x � 2 sin2x + 43

. 4 sin2x . cos2x = 0

1 � 2 sin2x + sin2x cos2x = 0 sin4x + sin2x � 1 = 0

sin2x = 2

15 cos 2x = 2 � 5

x = n ± 21

cos�1 (2 � 5 ), n

alp solutions trigonometric ratio & identities maths hindi

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