522 CHAPTER 7 Analytic Trigonometry 60. α β α β α β ( ) ( ) + + − = cos cos 2 cos cos 61. α β α β α β ( ) + = + sin sin cos 1 cot tan 62. α β α β α β ( ) + = + sin cos cos tan tan 63. α β α β α β ( ) + = − cos cos cos 1 tan tan 64. α β α β α β ( ) − = + cos sin cos cot tan 65. α β α β α β α β ( ) ( ) + − = + − sin sin tan tan tan tan 66. α β α β α β α β ( ) ( ) + − = − + cos cos 1 tan tan 1 tan tan 67. α β α β β α ( ) + = − + cot cot cot 1 cot cot 68. α β α β β α ( ) − = + − cot cot cot 1 cot cot 69. α β α β α β ( ) + = − sec csc csc cot cot 1 70. α β α β α β ( ) − = + sec sec sec 1 tan tan 71. α β α β α β ( ) ( ) − + = − sin sin sin sin 2 2 72. α β α β α β ( ) ( ) − + = − cos cos cos sin 2 2 73. θ π θ ( ) ( ) + = − k sin 1 sin , k k any integer 74. θ π θ ( ) ( ) + = − k cos 1 cos , k k any integer In Problems 75–86, find the exact value of each expression. 75. ( ) + − − sin sin 1 2 cos 0 1 1 76. + ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ − − sin sin 3 2 cos 1 1 1 77. ( ) ⎡ − − ⎢ ⎣ ⎤ ⎦ ⎥ − − sin sin 3 5 cos 4 5 1 1 78. ( ) ⎡ − − ⎣ ⎢ ⎤ ⎦ ⎥ − − sin sin 4 5 tan 3 4 1 1 79. ( ) + − − cos tan 4 3 cos 5 13 1 1 80. ( ) − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − − cos tan 5 12 sin 3 5 1 1 81. ( ) − − − cos sin 5 13 tan 3 4 1 1 82. ( ) + − − cos tan 4 3 cos 12 13 1 1 83. π ( ) + − tan sin 3 5 6 1 84. π( ) − − tan 4 cos 3 5 1 85. ( ) + − − tan sin 4 5 cos 1 1 1 86. ( ) + − − tan cos 4 5 sin 1 1 1 In Problems 87–92, write each trigonometric expression as an algebraic expression containing u and v. Give the restrictions required on u and v. 87. ( ) + − − u v cos cos sin 1 1 88. ( ) − − − u v sin sin cos 1 1 89. ( ) − − − u v sin tan sin 1 1 90. ( ) + − − u v cos tan tan 1 1 91. ( ) − − − u v tan sin cos 1 1 92. ( ) + − − u v sec tan cos 1 1 In Problems 93–98, solve each equation on the interval θ π ≤ < 0 2 . 93. θ θ − = sin 3 cos 1 94. θ θ + = 3 sin cos 1 95. θ θ + = sin cos 2 96. θ θ − = − sin cos 2 97. θ θ + = tan 3 sec 98. θ θ + = − cot csc 3 Applications and Extensions 99. Show that ( ) + = − − v v sin sin cos 1. 1 1 100. Show that ( ) + = − − v v cos sin cos 0. 1 1 101. Calculus Show that the difference quotient for = f x x ( ) sin is given by ( ) + − = + − = ⋅ − ⋅ − f x h f x h x h x h x h h x h h ( ) ( ) sin sin cos sin sin 1 cos 102. Calculus Show that the difference quotient for = f x x ( ) cos is given by ( ) + − = + − =− ⋅ − ⋅ − f x h f x h x h x h x h h x h h ( ) ( ) cos cos sin sin cos 1 cos 103. One, Two, Three (a) Show that ( ) + + = − − − tan tan 1 tan 2 tan 3 0. 1 1 1 (b) Conclude from part (a) that π + + = − − − tan 1 tan 2 tan 3 1 1 1 Source: College Mathematics Journal, Vol. 37, No. 3, May 2006 104. Electric Power In an alternating current (ac) circuit, the instantaneous power p at time t is given by φ ω φ ω ω ( ) ( ) ( ) ( ) = − p t V I t V I t t cos sin sin sin cos m m m m 2 Show that this is equivalent to ω ω φ ( ) ( ) ( ) = − p t V I t t sin sin m m Source: HyperPhysics, hosted by Georgia State University
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