722 CHAPTER 7 Trigonometric Identities and Equations 62. cot u 2, given tan u = - 25 2 , with 90° 6u 6180° 63. sin u, given cos 2u = 3 5 and u terminates in quadrant I 64. cos u, given cos 2u = 1 2 and u terminates in quadrant II 65. cos x, given cos 2x = - 5 12, with p 2 6x 6p 66. sin x, given cos 2x = 2 3, with p6x 6 3p 2 Verify that each equation is an identity. See Example 12. 79. 1sin x + cos x22 = sin 2x + 1 80. sec 2x = sec2 x + sec4 x 2 + sec2 x - sec4 x 81. 1cos 2x + sin 2x22 = 1 + sin 4x 82. 1cos 2x - sin 2x22 = 1 - sin 4x 83. tan 8u - tan 8u tan2 4u = 2 tan 4u 84. sin 2x = 2 tan x 1 + tan2 x 85. cos 2u = 2 - sec2 u sec2 u 86. tan 2u = -2 tan u sec2 u - 2 87. sin 4x = 4 sin x cos x cos 2x 88. 1 + cos 2x sin 2x = cot x 89. 2 cos 2u sin 2u = cot u - tan u 90. cot 4u = 1 - tan2 2u 2 tan 2u 91. sec2 x 2 = 2 1 + cos x 92. cot2 x 2 = 11 + cos x22 sin2 x 93. sin2 x 2 = tan x - sin x 2 tan x 94. sin 2x 2 sin x = cos2 x 2 - sin2 x 2 95. 2 1 + cos x - tan2 x 2 = 1 96. tan u 2 = csc u - cot u 97. 1 - tan2 u 2 = 2 cos u 1 + cos u 98. cos x = 1 - tan2 x 2 1 + tan2 x 2 Simplify each expression. See Example 11. 67. B1 - cos 40° 2 68. B1 + cos 76° 2 69. B1 - cos 147° 1 + cos 147° 70. B1 + cos 165° 1 - cos 165° 71. 1 - cos 59.74° sin 59.74° 72. sin 158.2° 1 + cos 158.2° 73. {B1 + cos 18x 2 74. {B1 + cos 20a 2 75. {B1 - cos 8u 1 + cos 8u 76. {B1 - cos 5A 1 + cos 5A 77. {B1 + cos x 4 2 78. {D1 - cos 3u 5 2 Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically. 99. cos4 x - sin4 x 100. 4 tan x cos2 x - 2 tan x 1 - tan2 x 101. 2 tan x 2 - sec2 x 102. cot2 x - 1 2 cot x
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