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42. If tan a = m, and tan ß= n, prove that

=

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43. If tan0 = (a + 1), and tan $ = (a – 1), prove that

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=

44. cos (2 — Y+z)= cos x cos y cos 2 + cos x sin y sinz

- sin x cos y sinz + sin æ sin y cos z.

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45. sin (2 Y - 2) = sin x + sin y + sinz

+ 4 sin } (x y) sin } (x— z) sin }(y+z).

46. sin (ac + y 2) + sin(x +%y) + sin (y +2 — x)

= sin(x+y+z) + 4 sin x sin y sin z.

47. sin 2:0 + sin 2 y + sin 2z - sin 2 (x + y + x)

+ z = 4 sin (x + y) sin (y +z) sin (z + x).

48. cos 2 x + cos2 y + cos2z + cos 2 (x + y + x)

= 4 cos (2 + y) cos (y + 2) cos (2 + x).

49. cos (x + y - x) + cos (y +2 — 2) + cos (x + x - y)

+cos (x + y + x) = 4 cos x cos y cos 2.

50. sin’x + sinoy+sin’z + sino (x + y + 2)

= 21- cos (x + y) cos (y+z) cos (z+x)}.

51. cosa x + cos2y + cos2z + cos2 (x + y - x)

=2{1+ cos (x + y) cos (2—2) cos (y-2)}. 52. cosx sin(y-2) + cos y sin (2— 2) + cos z sin(a —

- y)=0. 53. sin x sin(y 2) + sin y sin (2-2) + sinz sin(x - y)= 0.

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54. cos (a + y) cos (x - y) + sin (y + 2) sin (y-)

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72. tan (45° + x) – tan (45° — 2)=2 tan 2x.

73. tan (45° — 2) + cot (45° - 2) = 2 sec 2 x.

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tan?(45° + x) – 1 74.

tanʼ(45° + 2) +1 75. cos (x + 45°)

= sec 2 x – tan 2 x. cos (x – 45°) 76. tan x =

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sin x + sin 2x 1 + cos x + cos 2 x

77.

tan x =

sin 2 x — sin x 1 COS 2 + cos 2 x

78.

cos 3x

2 cos 2 x - 1.

COS C

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84. sin 2 x sin 2 y=sin?(x + y) - sin?(x - y).

=

85. tan 50° + cot 50°= 2 sec 10°.

86. sin 3 x = 4 sin x sin (60° + x) sin (60° — «).

=

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89. 2 cos

so 2 + V2. 90. (3sin 6 — 4 sin?)? + (4 cos 6 - 3 cos 6)? = 1.

2

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0

0 93. Prove that tan and cot

are the roots of the

2 equation

22 — 2 x cosec 0 +1=0.

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95. Find the values of (1) sin 9°, (2) cos 9°, (3) sin 81°, (4) cos 1890, (5) tan 2021, (6) tan 97].

Ans. (1) I (V3+ V5 – V5 – V5),

(2) }(V3+ V5 + V5 – V5),
(3) sin 81°= cos 90,

,
(4) cos 189°= -cos 9o,
(5) 12-1,

(6) -(V3+ V2)(V2 +1). 96. If A = 200°, prove that

A
1)
(1) 2 sin + V1 + sin A + V1 – sin A.

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97. If A lies between 270° and 360°, prove

that

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98. If A lies between 450° and 630°, prove that

A

=-11+ sin A - V1 - sin A. 2

(1) 2 sin

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