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16. sin 2 x + sin 4 x = 2 sin 3 x cos x.
17. sin 4 x = 4 sin x cos X . 8 sinox Cos X.
18. sin 4 x = 8 cos*x sin a 4 cos x sin x.
19. cos 4 x = 1- 8 cos” x + 8 cos4 x =1-8 sin’x + 8 sin* x.
20. cos 2 x + cos 4 x 2 cos 3 x cos x.
21. sin 3 x sin x = 2 cos 2 x sin x.

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Prove the following identities :

28. (sin 2 x – sin 2 y) tan(x + y)= 2 (siner -- sinäy). x 29. sin 3 x = 4 sin x sin (60° + x) sin (60° — x). 30. sin 4 x = 2 sin x cos 3x + sin 2 x.

31. sin x + sin(x - ) + sin( - )= 0. 32. cos x sin(y – z)+cos y sin (z — x) + cos z sin (x – y)=0. 33. cos (x + y) sin y - cos (x + x) sin z

= sin(x + y) cos y - sin(x + 2) cos z. 34, cos (x+y+z) + cos (x + y - x) + cos (x y + x)

+ cos (y + 2 = 4 cos x cos y cos z. 35. sin (x + y) cos(x - y) + sin(y + x) cos(y z)

+ sin(x + x) cos(z — x)= sin 2 x + sin 2 y + sin 2 z. 36. sin (x + y) + cos (x - y)=2 sin (x + ) sin(y + 7). 37. sin(x + y) – cos (x – y)=– 2 sin (x – 7) sin(y – }). 38. cos (x + y) cos(x - y)= cos x – sino y. 39. sin(x + y) sin(x - y)=sin’ x — sin” y. 40. sin x + 2 sin 3 x + sin 5 x = 4 cos2x sin 3 x.

2) =

If A, B, C are the angles of a triangle, prove that : 41. sin 2 A + sin 2 B + sin 2C : 4 sin A sin B sin C. 42. cos 2 A + cos 2 B + cos 2 C = 1 4 cos A cos B cos C. 43. sin 3 A + sin 3B + sin 30 4 cos į A cos zB cos C. 44. cos A + cos”B + cosC =1 – 2 cos A cos B cos C.

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If A + B + C = 90°, prove that:
45. tan A tan B + tan B tan C + tan C tan A = 1.
46. sin’A + sin’B + sino C 1- 2 sin A sin B sin C.
47. sin 2 A + sin 2 B + sin 2C = 4 cos A cos B cos C.
48. Prove that cot-13 + esc-1 V5 = 4 .
49. Prove that x + tan-'(cot 2 x) = tan-1 (cot x).

Prove the following statements :

sin 75° + sin 15° 50.

tan 60°. sin 75° · sin 15° 51. sin 60° + sin 120° = 2 sin 90° cos 30°. 52. cos 20° + cos 100° + cos 140o = 0. 53. cos 36° + sin 36° v2 54. tan 11° 15' + 2 tan 22° 30' + 4 tan 45o = cot 11° 15'.

cos 9o.

149. How to solve a Trigonometric Equation. To solve a trigonometric equation is to find for the unknown quantity the general value which satisfies the equation.

Practically it suffices to find the values between 0° and 360°, since we can then apply our knowledge of the periodicity of the various functions to give us the other values if we need them,

There is no general method applicable to all cases, but the following suggestions will prove of value:

1. If functions of the sum or difference of two angles are involved, reduce such functions to functions of a single angle.

Thus, instead of leaving sin (x + y) in an equation, substitute for sin (x + y) its equal sin x cos y + cos x sin y.

Similarly, replace cos 2 x by cos? x — sina &, and replace the functions of fic by the functions of x.

2. If several functions are involved, reduce them to the same function.

This is not always convenient, but it is frequently possible to reduce the equation so as to involve only sines and cosines, or tangents and cotangents, after which the solution can be seen.

3. If possible, employ the method of factoring in solving the final equation.

4. Check the results by substituting in the given equation.
For example, solve the equation cos x = sin 2x.
By $ 101,

sin 2x = 2 sin x cos X.

... COS X = 2 sin x cos X. .: (1 – 2 sin x) cos x = 0.

.. cos x = 0, or 1 - 2 sin x = 0. .. X = 90° or 270°, 30° or 150°, or these values increased by 2 no. Each of these values satisfies the given equation.

Exercise 77. Trigonometric Equations

7. sin x = cos 2 x.

8. tan x tan 2x = 2.

Solve the following equations :
1. sin x = 2 sin(fn + x).
2. sin 2x = 2 cos x.
3. cos 2 x = 2 sin x.
4. sin x + cos x =

1.
5. sin x + cos 2 x = 4 sinox.
6. 4 cos 2 x + 3 cos x =

= 1.

9. secx = 4 csc x.
10. cos A + cos 20 = 0.
11. cot 1 0 + csc 0 2.
12. cotx tan 2 x = 3.

cos X.

cosa x

Solve the following equations : 13. sin x + sin 2 x = sin 3 x. 33. sin x sec 2 x = 1. 14. sin 2x = 3 sinox

34. sin’x + sin 2 = 1. 15. cotA ftan 0.

35. cos x sin 2 x csc X = 1. 16. 2 sin A = cos 0.

36. cotx tan 2 x = sec 2 x. 17. 2 sin x + 5 sin x = 3.

37. sin 2 x = cos 4 x. 18. tan x sec x = V2.

38. sin 2 z cot z sin? = 19. COS X cos 2 x - 1.

39. tan” x = sin 2 x. 20. cos 3 x + 8 cos8x = : 0.

40. sec 2 x +1

2 cos x. 21. tan x + cotx = tan 2 x.

41. tan 2 x + tan 3x = 0. 22. tan x + sec X = a.

42. csc x = 'cot x + 13. 23. cos 2 x = a (1 – cos x).

43. tan x tan 3x = - . 24. sin-'1x = 30°.

44. cos 5 x + cos 3 x + cos x = 0 25. tan- x + 2 cot-'x = : 135o. 45. sino x

k. 26. sec x cotx = CSC X tan x. 46. sin x + 2 cos x = 1. 27. tan 2 x tan x = 1.

47. sin 4 x cos 3 x = sin 2 x. 28. tan x + cotx = .

48. sin x + cos x = sec X. 29. sin x + sin 2 x 1- cos 2 x. 49. 2 cos x cos 3x+1= 0. 30. 4 cos 2 x + 6 sin x = 5.

50. cos 3x 2 cos 2 x + cos x =( 31. sin 4x sin 2x = sin x. 51. sin (x – 30°)= { V3 sin x. 32. 2 sino a + sino 2 x = 2.

52. sin- x + 2 cos-2x = . 53. sin-1x + 3 cos-2x = 210°.

1 · tan x
54.

= cos 2 x.
1+ tan x
55. tan(7 + + x) + tan($ 7 – x)= 4.
56. V1 + sin x V1 – sin x = 2 cos X.
57. sin (45° + x) + cos (45° — x)= 1.
58. (1 – tan x) cos 2 x = a (1 + tan x).
59. sino x + cosRx isin2 x.
60. sec (x + 120°) + sec (x – 120°)= 2 cos x.
61. sinox cos x cos’x – sin’x +1= 0.
62. sin x + sin 2 x + sin 3 x = 0.
63. sin A + 2 sin 20 + 3 sin 30 = 0.
64. sin 3 x = cos 2 x - 1.
65. sin(x +12°) + sin(x – 8°)= sin 20°.

cos* x =

7 2 5

Solve the following equations :

66. tan (60° + x) tan (60° — x)=– 2.
67. tan x + tan 2 x = = 0.
68. sin(x +120°) + sin(x + 60°)= i.
69. sin(x +30°) sin(x - 30°)= 1.
70. sin 20

= cos 30.
71. sin* x + cos*x =
72. sino a
73. tan (x + 30°)= 2 cos x.
74. sec x = 2 tan x + 1
75. sin 11 x sin 4x + sin 5 x sin 2 x = 0.
76. cos x + cos 3 x + cos 5 x + cos 7 x = 0.
77. sin (x +12) cos (x – 12°) = cos 33° sin 57
78. sin-1x + sin-1 { x = 120°.
79. tan-1x + tan-12x = tan-133.
80. tan-'(x+1)+ tan-'(x - 1)= tan- 2x.
81. (3 – 4 cos x) sin 2 x = 0.
82. cos 2 0 sec 0 + sec 0 +1= 0.
83. sin x cos 2 x tan x cot 2 x secx csc 2 x =1.
84. tan (0 + 45°)= 8 tan 0.
85. tan (0 + 45°) tan 0 :
86. sin u + sin 3x = COS X cos 3 x.
87. sin £ x (cos 2 a 2)(1 – tanox)= 0.
88. tan x + tan 2 x = tan 3 x.
89. cotx tan x = sin x + cos X.

= 2.

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secx

CSCX

csca

sec- x

Prove the following identities :
90. (1 + cotx + tan x) (sin x cos x)=
91. 2 csc 2 x cot x =1+ cotx.
92. sin a + sin b + sin(a + b)= 4 cos } a cos } b sin }(a + b).
93. tan (45° + x) – tan (45° — x)= 2 tan 2 x.
94. cotax cosx = cotx cosx.
95. tanox

tanox sinox.
96. cot* x + cot” x cscfx cscʻx.
97. cos x + sin’x cos'y = cosy +sin'y cosas.

sin x =

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