Marks Booster

Day – 1
1. Prove the following identities:
(i) sin8 𝜃 − cos 8 𝜃 = (sin2 𝜃 − cos 2 𝜃)(1 − 2 sin2 𝜃 cos 2 𝜃)
2. Prove the following identities:
(a) (1 + cot 𝜃 − 𝑐𝑜𝑠𝑒𝑐 𝜃)(1 + tan 𝜃 + sec 𝜃) = 2
tan 𝜃+sec 𝜃−1 1+sin 𝜃
(b) =
tan 𝜃−sec 𝜃+1 cos 𝜃
3. If cos 𝜃 + sin 𝜃 = √2 cos 𝜃, prove that cos 𝜃 − sin 𝜃 = √2 sin 𝜃.
4. If 10𝑠𝑖𝑛4 𝛼 + 15𝑐𝑜𝑠 4 𝛼 = 6, find the value of 27 𝑐𝑜𝑠𝑒𝑐 6 𝛼 + 8𝑠𝑒𝑐 6 𝛼.
5. Prove that: Sin6 𝜃 + cos6 𝜃 = 1 – 3 sin2 𝜃 cos2 𝜃
𝑠𝑖𝑛 3 𝐴+𝑐𝑜𝑠3 𝐴 𝑠𝑖𝑛3 𝐴−𝑐𝑜𝑠3 𝐴
6. Prove that: + =2
sin 𝐴+cos 𝐴 sin 𝐴−cos 𝐴
𝑡𝑎𝑛3 𝜃 𝑐𝑜𝑡 3 𝜃 1−2𝑠𝑖𝑛 2 𝜃 𝑐𝑜𝑠2 𝜃
7. Prove that: 1+𝑡𝑎𝑛2𝜃 + 1+𝑐𝑜𝑡 2𝜃 = sin 𝜃 cos 𝜃

Day – 2
8. Find the values of the following trigonometric ratios:
(a) cos(−4800)
(b) sin(−11250)
9. Find the values of the following trigonometric ratios.
(a) tan 4800
−15𝜋
(b) cot ( )
4
10. Prove that: cos 5100 cos 3300 + sin 3900 cos 1200 = -1 .
11. Prove that: sin (-4200) (cos 3900) + cos (-6600) sin 3300) = -1
12. In any quadrilateral ABCD, prove that
(a) sin(𝐴 + 𝐵) + sin(𝐶 + 𝐷) = 0
13. Find the value of the expression
3𝜋 𝜋
3 {𝑠𝑖𝑛4 ( 2 − 𝜃) + 𝑠𝑖𝑛4 (3𝜋 + 𝜃)} − 2 {𝑠𝑖𝑛6 ( 2 + 𝜃) + 𝑠𝑖𝑛 6 (5𝜋 − 𝜃)}
14. Prove that:
𝜋 𝜋
(a) {1 + cot 𝜃 − sec ( 2 + 𝜃)} {1 + cot 𝜃 + sec ( 2 + 𝜃)} = 2 cot 𝜃
15. In a ∆𝐴𝐵𝐶, 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡:
(a) cos(𝐴 + 𝐵) + cos 𝐶 = 0
𝐴+𝐵 𝐶
(b) cos ( ) = sin 2
2
𝐴+𝐵 𝐶
(c) tan = cot
2 2
16. If A, B, C, D be the angles of a cyclic quadrilateral, taken in order, prove that:
cos(1800 − 𝐴) + cos(1800 + 𝐵) + cos(1800 + 𝐶) − sin(900 + 𝐷) = 0
17. Prove that:
1
(a) sin 7800 sin 4800 + cos 1200 sin 1500 = 2
(b) tan 2250 cot 4050 + tan 7650 cot 6750 = 0

18. If sin 𝑥 + 𝑐𝑜𝑠𝑒𝑐 𝑥 = 2, then write the value of 𝑠𝑖𝑛 𝑛 𝑥 + 𝑐𝑜𝑠𝑒𝑐 𝑛 𝑥.

19. If sin 𝑥 + 𝑠𝑖𝑛 2 𝑥 = 1, then write the value of 𝑐𝑜𝑠12 𝑥 + 3 𝑐𝑜𝑠10 𝑥 + 3𝑐𝑜𝑠 8 𝑥 + 𝑐𝑜𝑠 6 𝑥.
20. Write the value of 2 (𝑠𝑖𝑛 6𝜃 + 𝑐𝑜𝑠 6 𝜃) − (𝑠𝑖𝑛4 𝜃 + 𝑐𝑜𝑠 4 𝜃) + 1.

Day – 3
21. Prove that: tan 750 + cot 750 = 4.
22. If 3 tan 𝐴 tan 𝐵 = 1, prove that 2 cos(𝐴 + 𝐵) = cos(𝐴 − 𝐵).
sin(𝑥+𝑦) tan 𝑥 +tan 𝑦
23. Prove that: sin(𝑥−𝑦) = tan 𝑥 −tan 𝑦
sin(𝐵−𝐶) sin(𝐶−𝐴) sin(𝐴−𝐵)
24. Prove that: cos 𝐵 cos 𝐶 + cos 𝐶 cos 𝐴 + cos 𝐴 cos 𝐵 = 0
25. If sin 𝐵 = 3 sin(2𝐴 + 𝐵), prove that 2 𝑡𝑎𝑛 𝐴 + tan(𝐴 + 𝐵) = 0
26. If 2 tan 𝛽 + cot 𝛽 = tan 𝛼, prove that cot 𝛽 = 2 tan(𝛼 − 𝛽).
𝑛 sin 𝛼 cos 𝛼
27. If tan 𝛽 = , show that tan(𝛼 − 𝛽) = (1 − 𝑛) tan 𝛼.
1−𝑛 𝑠𝑖𝑛2 𝛼
28. Prove that:
(a) tan 3𝐴 tan 2𝐴 tan 𝐴 = tan 3𝐴 − tan 2𝐴 − tan 𝐴
(b) cot 𝐴 cot 2𝐴 − cot 2𝐴 cot 3𝐴 − cot 3𝐴 cot 𝐴 = 1
𝑚 1 𝜋
29. If 𝛼 𝑎𝑛𝑑 𝛽 are acute angles such that tan 𝛼 = 𝑚+1 𝑎𝑛𝑑 tan 𝛽 = 2𝑚+1, prove that 𝛼 + 𝛽 = 4 .
𝜋
30. If 𝐴 + 𝐵 = 4 , prove that:
(a) (1 + tan 𝐴)(1 + tan 𝐵 ) = 2
(b) (cot 𝐴 − 1) (cot 𝐵 − 1) = 2
𝜋 𝐴 𝜋 𝐴 1
31. Prove that: 𝑠𝑖𝑛 2 ( 8 + 2 ) − 𝑠𝑖𝑛2 ( 8 − 2 ) = sin 𝐴.
√2
4 5 𝜋 56
32. If cos(𝛼 + 𝛽) = 5 , sin(𝛼 − 𝛽) = 13 𝑎𝑛𝑑 𝛼, 𝛽 lie between 0 and 4 , 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 tan 2𝛼 = 33.
33. Prove that: tan 700 = tan 200 + 2 tan 500.
34. If tan(𝛼 + 𝜃) = 𝑛 tan(𝛼 − 𝜃) , show that: (𝑛 + 1) sin 2𝜃 = (𝑛 − 1) sin 2𝛼.
1 1
35. If tan 𝐴 − tan 𝐵 = 𝑥 and cot 𝐵 − cot 𝐴 = 𝑦, prove that cot(𝐴 − 𝐵) = 𝑥 + 𝑦
𝜋 1
36. If tan(𝜋 cos 𝜃) = cot(𝜋 sin 𝜃), prove that cos (𝜃 − ) = ±
4 2√2
37. If sin 𝛼 + sin 𝛽 = 𝑎 𝑎𝑛𝑑 cos 𝛼 + cos 𝛽 = 𝑏, show that
𝑏2 −𝑎2
(a) cos(𝛼 + 𝛽) = 𝑏2+𝑎2
2𝑎𝑏
(b) sin(𝛼 + 𝛽) = 𝑎2+𝑏2
1 √3 𝜋 𝜋
38. If sin 𝐴 = 2 , cos 𝐵 = , 𝑤ℎ𝑒𝑟𝑒 2 < 𝐴 < 𝜋 𝑎𝑛𝑑 0 < 𝐵 < 2 , find the following:
2
(a) tan(𝐴 + 𝐵) 0
(b) tan(𝐴 − 𝐵) −√3
39. Prove that:
cos 110 +sin 110
(a) = tan 650
cos 110 −sin 110
cos 90 +sin 90
(b) = tan 540
cos 90 −sin 90
cos 80 0 sin 80
(c) = tan 370
cos 80 +sin 80
tan 690 +tan 660
40. Prove that: 1−tan 690 tan 660 = −1

41.
5 𝜋
(a) If 𝑡𝑛𝑎 𝐴 = 6 𝑎𝑛𝑑 tan 𝐵 = 1/11, prove that 𝐴 + 𝐵 = 4
𝑚 1 𝜋
(b) If tan 𝐴 = 𝑚−1 𝑎𝑛𝑑 tan 𝐵 = 2𝑚−1, then prove that 𝐴 − 𝐵 = 4
sin(𝐴−𝐵) sin(𝐵−𝐶) sin(𝐶−𝐴)
42. Prove that: + cos 𝐵 cos 𝐶 + cos 𝐶 cos 𝐴 = 0
cos 𝐴 cos 𝐵
1 1
43. If tan 𝐴 + tan 𝐵 = 𝑎 𝑎𝑛𝑑 cot 𝐴 + cot 𝐵 = 𝑏, 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡: cot(𝐴 + 𝐵) = 𝑎 − 𝑏 .
sin(𝐴−𝐵) 𝑥−1
44. If tan 𝐴 = 𝑥 tan 𝐵 , 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 sin(𝐴+𝐵) = 𝑥+1
1 𝜋
45. If sin(𝛼 + 𝛽) = 1 𝑎𝑛𝑑 sin(𝛼 − 𝛽) = 2, where 0 ≤ 𝛼, 𝛽 ≤ 2 , then find the values of tan(𝛼 + 2𝛽)
and tan(2𝛼 + 𝛽).
sin 𝛼−cos 𝛼
46. 𝑖𝑓 tan 𝜃 = sin 𝛼+cos 𝛼, then show that sin 𝛼 + cos 𝛼 = √2 cos 𝜃
sin(𝑥+𝑦) 𝑎+𝑏 tan 𝑥 𝑎
47. If sin(𝑥−𝑦) = 𝑎−𝑏 , 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 tan 𝑦 = 𝑏 .
𝑝+𝑞
48. If tan(𝐴 + 𝐵) = 𝑝, tan(𝐴 − 𝐵) = 𝑞, 𝑡ℎ𝑒𝑛 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 tan 2𝐴 = .
1−𝑝𝑞
𝜋
49. Prove that 5cos 𝜃 + 3 cos (𝜃 + ) + 3 lie between – 4 and 10.
3
1
50. Prove that: cos 20 cos 40 cos 600 cos 800 = 16
0 0

1
51. Prove that: sin 100 sin 300 sin 500 sin 700 = 16
3
52. Prove that: sin 200 sin 400 sin 600 sin 800 = 16
53. Prove that: tan 200 tan 400 tan 800 = tan 600
1
54. Prove that: sin 𝐴 sin(600 − 𝐴) sin(600 + 𝐴) = sin 3𝐴
4
 𝜋 2𝜋
55. Prove that: 4 sin 𝜃 sin ( + 𝜃) sin ( + 𝜃) = sin 3𝜃
3 3
56. Prove that:
3
(a) cos 100 cos 300 cos 500 cos 700 = 16
√3
(b) sin 100 sin 500 sin 600 sin 700 = 16
3
(c) sin 200 sin 400 sin 600 sin 800 =
16
57. Prove that: tan 𝜃 (tan 600 − 𝜃) tan(60 + 𝜃) = tan 3𝜃. 0

58. Prove that:

cos 2𝐴 cos 3𝐴 − 2 cos 7𝐴 cos 2𝐴 + 2 cos 10𝐴 cos 𝐴
= cot 6𝐴 cot 5𝐴
sin 4𝐴 sin 3𝐴 − sin 2𝐴 sin 5𝐴 + sin 4𝐴 sin 7𝐴
sin(𝜃+𝛼) 1−𝑚 𝜋 𝜋
59. If = , prove that tan ( − 𝜃) tan ( − 𝛼) = 𝑚.
cos(𝜃−𝛼) 1+𝑚 4 4
60.
tan(𝐴+𝐵) 𝜆+1
(a) If sin 2𝐴 = 𝜆 sin 2𝐵 , 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡: tan(𝐴−𝐵) = 𝜆−1
𝑚+𝑛
(b) If m sin 𝜃 = 𝑛 sin(𝜃 + 2𝛼), prove that tan(𝜃 + 𝛼) = 𝑚−𝑎 tan 𝛼
61. If cos(𝐴 + 𝐵) sin(𝐶 − 𝐷) = cos(𝐴 − 𝐵) sin(𝐶 + 𝐷) , 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡
tan 𝐴 tan 𝐵 tan 𝐶 + tan 𝐷 = 0
2𝜋 4𝜋
62. If 𝑥 cos 𝜃 = 𝑦𝑐𝑜𝑠 (𝜃 + ) = 𝑧 cos (𝜃 + ) , prove that 𝑥𝑦 + 𝑦𝑧 + 𝑧𝑥 = 0
3 3

𝜋
63. Show that: √2 + √2 + √2 + 2 cos 8𝜃 = 2 cos 𝜃 , 0 < 𝜃 < 8
sec 8𝜃−1 tan 8𝜃
64. Prove that: =
sec 4𝜃−1 tan 2𝜃
65. Prove that:
2𝜋 2𝜋 3
(a) 𝑐𝑜𝑠 2 𝐴 + 𝑐𝑜𝑠 2 (𝐴 + ) + 𝑐𝑜𝑠 2 (𝐴 − )=2
3 3
𝜋 𝜋 3
(b) 𝑐𝑜𝑠 2 𝐴 + 𝑐𝑜𝑠 2 (𝐴 + 3 ) + 𝑐𝑜𝑠 2 (𝐴 − 3 ) = 2
66. Show that √3 𝑐𝑜𝑠𝑒𝑐 200 − sec 200 = 4
sin 2𝑛 𝐴
67. Prove that: cos 𝐴 cos 2𝐴 cos 22 𝐴 cos 23 𝐴 … cos 2𝑛−1 𝐴 = 2𝑛 sin 𝐴
𝜃 1−𝑒 𝜙 cos 𝜃−𝑒
68. If tan = √ tan , prove that cos 𝜙 = .
2 1+𝑒 2 1−𝑒 cos 𝜃
69. Find the values of
𝜋
(a) cos 8
𝜋
(b) sin 8
𝜋
(c) tan 8
𝜋
(d) sin 24
𝜋
(e) cos 24
1
70. Prove that: cos 𝐴 cos(60 − 𝐴) cos(60 + 𝐴) = 4 cos 3𝐴.
 1
71. Prove that: sin 𝐴 sin(60 − 𝐴) sin(60 + 𝐴) = sin 3𝐴.
4
0 0 0 0 3
72. Prove that: sin 20 sin 40 sin 60 sin 80 = 16.
73. Prove that:
(a) tan 𝐴 + tan(600 + 𝐴) − tan(600 − 𝐴) = 3 tan 3𝐴
(b) cot 𝐴 + cot(600 + 𝐴) − cot(600 − 𝐴) = 3 cot 3𝐴
√5−1
74. Prove that: sin 180 = .
4
√10+2√5
75. Prove that: cos 180 = 4
√5+1
76. Prove that: cos 360 = 4
√10−2√5
77. Prove that: sin 360 = .
4

Day – 4
78. Solve the equation: sin 𝜃 + sin 3𝜃 + sin 5𝜃 = 0.
79. Solve the equation: cos 𝜃 + cos 3𝜃 − 2 cos 2𝜃 = 0.
80. Solve the following equations:
(a) 2 𝑐𝑜𝑠 2 𝜃 + 3 sin 𝜃 = 0
3
(b) 𝑐𝑜𝑡 2 𝜃 + sin 𝜃 + 3 = 0
(c) 2 tan 𝜃 − cot 𝜃 = −1
(d) 4 cos 𝜃 − 3 sec 𝜃 = tan 𝜃
(e) 𝑡𝑎𝑛 2 𝜃 + (1 − √3) tan 𝜃 − √3 = 0
(f) 𝑠𝑒𝑐 2 2𝑥 = 1 − tan 2𝑥

81. Solve the following equations:

(a) tan 𝜃 + tan 2𝜃 + tan 𝜃 tan 2𝜃 = 1
(b) tan 𝜃 + tan 2𝜃 + tan 3𝜃 = tan 𝜃 tan 2𝜃 tan 3𝜃
(c) tan 𝜃 + tan 2𝜃 + √3 tan 𝜃 tan 2𝜃 = √3
𝜋 2𝜋
(d) tan 𝜃 + tan (𝜃 + ) + tan (𝜃 + )=3
3 3
82. Solve: 2 𝑠𝑖𝑛 2𝑥 + 𝑠𝑖𝑛 2𝑥 = 2.
2

83. Solve: 4 sin 𝑥 sin 2𝑥 sin 4𝑥 = sin 3𝑥.

84. Solve: √3 cos 𝜃 + sin 𝜃 = √2.
85. Solve: √2 sec 𝜃 + tan 𝜃 = 1.
86. Solve: cot 𝜃 + 𝑐𝑜𝑠𝑒𝑐 𝜃 = √3.
 Answers
4. 250 18. 2

𝟏 1 19. 1
8. (a) − (b) −
𝟐 √2
20. 0
9. (a) −√3 (b) 1
1
45. −√3, −
13. 1 √3

√𝟐+𝟏 √2−1 4−√6−√2 4+√6+√2

69. (a) √ (b) √ (c) √2 − 1 (d) √ (e) √
𝟐√𝟐 2√2 2√2 2√2

𝑛𝜋 𝜋
78. 𝑜𝑟 𝑚𝜋 ± 3 , 𝑤ℎ𝑒𝑟𝑒 𝑚 ∈ 𝑍
3

79. 2𝑚𝜋, 𝑤ℎ𝑒𝑟𝑒 𝑚, 𝑛 ∈ 𝑍

80.
𝜋
(a) 𝑛𝜋 + (−1)𝑛+1 , 𝑛 ∈ 𝑍 (d) 𝑛𝜋 + (−1)𝑛 𝛽 𝑤ℎ𝑒𝑟𝑒 𝑠𝑖𝑛 𝛽 =
−1−√17
6 8
𝜋
(b) 𝑚𝜋 + (−1)𝑚+1 2 , 𝑚, 𝑛 ∈ 𝑍 𝜋
(e) 𝑛𝜋 − 4 𝑜𝑟, 𝑚𝜋 + 3 , 𝑤ℎ𝑒𝑟𝑒 𝑚, 𝑛 ∈ 𝑍
𝜋

1 𝑛𝜋 3𝜋
(c) 𝑚𝜋 + 𝛼, 𝑤ℎ𝑒𝑟𝑒 𝑚, 𝑛 ∈ 𝑍 𝑎𝑛𝑑 𝑡𝑎𝑛 𝛼 = 2 (f) + ,𝑛 ∈ 𝑍
2 8
𝑛𝜋 𝜋 𝑛𝜋 𝑛𝜋 𝜋 𝑛𝜋 𝜋
81. (a) + 12 , 𝑛 ∈ 𝑍 (b) ,𝑛 ∈ 𝑍 (c) + 9,𝑛 ∈ 𝑍 (d) + 12 , 𝑛 ∈ 𝑍
3 3 3 3
 𝜋
82. 𝑚𝜋 ± 4 , 𝑤ℎ𝑒𝑟𝑒 𝑚, 𝑛 ∈ 𝑍

𝜋
83. 𝑚𝜋 ± 3 , 𝑤ℎ𝑒𝑟𝑒 𝑚, 𝑛 ∈ 𝑍

𝜋
84. 2𝑛𝜋 − 12 , 𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ 𝑍

𝜋
85. 2𝑛𝜋 − 4 , 𝑛 ∈ 𝑍

𝝅
86. 𝟐𝒏𝝅 + , 𝒏 ∈ 𝒁
𝟑