696 CHAPTER 7 Trigonometric Identities and Equations 84. 1 - cos u 1 + cos u = 2 csc2 u - 2 csc u cot u - 1 85. 2 sinx + cos x22 + 12 cos x - sinx22 = 5 86. sin2 x11 + cot x2 + cos2 x11 - tanx2 + cot2 x = csc2 x 87. sec x - cos x + csc x - sinx - sinx tanx = cos x cot x 88. sin3 u + cos3 u = 1cos u + sinu211 - cos u sinu2 Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically. 89. 1sec u + tanu211 - sinu2 90. 1csc u + cot u21sec u - 12 91. cos u + 1 sinu + tanu 92. tanu sinu + cos u Graph the expressions on each side of the equals symbol to determine whether the equation might be an identity. (Note: Use a domain whose length is at least 2p.) If the equation looks like an identity, then verify it algebraically. See Example 1. 93. 2 + 5 cos x sinx = 2 csc x + 5 cot x 94. 1 + cot2 x = sec2 x sec2 x - 1 95. tanx - cot x tanx + cot x = 2 sin2 x 96. 1 1 + sinx + 1 1 - sinx = sec2 x Show that the equation is not an identity by substituting a number for t. 97. sin1csc t2 = 1 98. 2cos2 t = cos t 99. csc t = 21 + cot2 t 100. cos t = 21 - sin2 t (Modeling) Work each problem. 101. Intensity of a Lamp According to Lambert’s law, the intensity of light from a single source on a flat surface at point P is given by I = k cos2 u, where k is a constant. (Data from Winter, C., Solar Power Plants, Springer-Verlag.) (a) Write I in terms of the sine function. (b) Why does the maximum value of I occur when u = 0? 102. Oscillating Spring The distance or displacement y of a weight attached to an oscillating spring from its natural position is modeled by y = 4 cos 2pt, where t is time in seconds. Potential energy is the energy of position and is given by P = ky2, where k is a constant. The weight has the greatest potential energy when the spring is stretched the most. (Data from Weidner, R., and R. Sells, Elementary Classical Physics, Vol. 2, Allyn & Bacon.) (a) Write an expression for P that involves the cosine function. (b) Use a fundamental identity to write P in terms of sin 2pt. P u y
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