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452 6 Trigonometric Identities and Conditional Equations Trigonometric functions are widely used in solving real-world problems as well as in the development of mathematics. Whatever their use, it is often of value to be able to change a trigonometric expression from one form to an equivalent more useful form. This involves the use of identities. Recall that an equation in one or more variables is said to be an identity if the left side is equal to the right side for all replacements of the variables for which both sides are defined. For example, the equation x2 2x 8 (x 4)(x 2) is an identity, but x2 2x 8 0 is not. The latter is called a conditional equation, since it holds only for certain values of x and not for all values for which both sides are defined. The first four sections of the chapter deal with trigonometric identities and the last section with conditional trigonometric equations. SECTION 6-1 Basic Identities and Their Use • Basic Identities • Establishing Other Identities In this section we review the basic identities introduced in Section 5-2 and show how they are used to establish other identities. • Basic Identities In the following box we list for convenient reference the basic identities introduced in Section 5-2. These identities will be used very frequently in the work that follows and should be memorized. Basic Trigonometric Identities Reciprocal Identities csc x 1 sin x sec x 1 cos x cot x Quotient Identities tan x sin x cos x cot x cos x sin x 1 tan x 6-1 Basic Identities and Their Use 453 Identities for Negatives sin (x) sin x cos (x) cos x tan (x) tan x Pythagorean Identities sin2 x cos2 x 1 tan2 x 1 sec2 x 1 cot2 x csc2 x All these identities were established in Section 5-2 (the second and third Pythagorean identities were established in Problems 87 and 88 in Exercise 5-2). EXPLORE-DISCUSS 1 • Establishing Other Identities EXAMPLE 1 Discuss an easy way to recall the second and third Pythagorean identities from the first. [Hint: Divide through the first Pythagorean identity by appropriate expressions.] Identities are established in order to convert one form to an equivalent form that may be more useful. To verify an identity means to prove that both sides of an equation are equal for all replacements of the variables for which both sides are defined. Such a proof might use basic identities or other verified identities and algebraic operations such as multiplication, factoring, combining and reducing fractions, and so on. The following examples illustrate some of the techniques used to verify certain identities. The steps illustrated are not necessarily unique—often, there is more than one path to a desired goal. To become proficient in the use of identities, it is important that you work out many problems on your own. Identity Verification Verify the identity: cos x tan x sin x Verification Generally, we proceed by starting with the more complicated of the two sides, and transform that side into the other side in one or more steps using basic identities, algebra, or other established identities. Thus, cos x tan x cos x sin x Matched Problem 1 EXAMPLE 2 Verify the identity: sin x cot x cos x Identity Verification Verify the identity: sec (x) sec x sin x cos x Quotient identity Algebra 454 6 Trigonometric Identities and Conditional Equations Verification sec (x) 1 cos (x) Reciprocal identity 1 cos x Identity for negatives sec x Matched Problem 2 EXAMPLE 3 Reciprocal identity Verify the identity: csc (x) csc x Identity Verification Verify the identity: cot x cos x sin x csc x Verification cos x cos x sin x sin x Quotient identity cos2 x sin x sin x Algebra cos2 x sin2 x sin x Algebra 1 sin x Pythagorean identity cot x cos x sin x csc x Reciprocal identity Key Algebraic Steps in Example 3 a a2 a2 b2 ab b b b b Matched Problem 3 Verify the identity: tan x sin x cos x sec x To verify an identity, proceed from one side to the other, or both sides to the middle, making sure all steps are reversible. Do not use properties of equality to perform the same operation on both sides of the equation. Even though no fixed method of verification works for all identities, certain steps help in many cases. Suggested Steps in Verifying Identities 1. Start with the more complicated side of the identity, and transform it into the simpler side. 6-1 Basic Identities and Their Use 455 2. Try algebraic operations such as multiplying, factoring, combining fractions, and splitting fractions. 3. If other steps fail, express each function in terms of sine and cosine functions, and then perform appropriate algebraic operations. 4. At each step, keep the other side of the identity in mind. This often reveals what you should do in order to get there. EXAMPLE 4 Identity Verification Verify the identity: Verification 1 sin x cos x 2 sec x cos x 1 sin x 1 sin x cos x (1 sin x)2 cos2 x cos x 1 sin x cos x (1 sin x) Algebra 1 2 sin x sin2 x cos2 x cos x (1 sin x) Algebra 1 2 sin x 1 cos x (1 sin x) Pythagorean identity 2 2 sin x cos x (1 sin x) Algebra 2(1 sin x) cos x (1 sin x) Algebra 2 cos x Algebra 2 sec x Reciprocal identity Key Algebraic Steps in Example 4 a b a2 b2 b a ba Matched Problem 4 EXAMPLE 5 Verify the identity: (1 c)2 1 2c c 2 1 cos x sin x 2 csc x sin x 1 cos x Identity Verification Verify the identity: sin2 x 2 sin x 1 1 sin x cos2 x 1 sin x m (a b) m n (a b) n 456 6 Trigonometric Identities and Conditional Equations Verification sin2 x 2 sin x 1 (sin x 1)2 cos2 x cos2 x Algebra (sin x 1)2 1 sin2 x Pythagorean identity (1 sin x)2 (1 sin x)(1 sin x) Algebra 1 sin x 1 sin x Algebra Key Algebraic Steps in Example 5 a 2 2a 1 (a 1)2 Matched Problem 5 EXAMPLE 6 Verify the identity: sec4 x 2 sec2 x tan2 x tan4 x 1 Identity Verification Verify the identity: Verification 1 b2 (1 b)(1 b) tan x cot x 1 2 cos2 x tan x cot x sin x cos x tan x cot x cos x sin x tan x cot x sin x cos x cos x sin x Change to sines and cosines (quotient identities). sin x cos x cos x sin x sin x cos x (sin x)(cos x) cos x sin x Multiply numerator and denominator by (sin x)(cos x), and use algebra to transform the compound fraction into a simple fraction. (sin x)(cos x) sin2 x cos2 x sin2 x cos2 x 1 cos2 x cos2 x 1 Pythagorean identity 1 2 cos2 x Algebra Key Algebraic Steps in Example 6 a b a b ab b a b a a 2 b2 2 a b a b2 a b ab b a b a 6-1 Matched Problem 6 Verify the identity: cot x tan x Basic Identities and Their Use 457 2 cos2 x 1 sin x cos x Just observing how others verify identities won’t make you good at it. You must verify a large number on your own. With practice the process will seem less complicated. EXAMPLE 7 Testing Identities Using a Graphing Utility Use a graphing utility to test whether each of the following is an identity. If an equation appears to be an identity, verify it. If the equation does not appear to be an identity, find a value of x for which both sides are defined but are not equal. (A) tan x 1 (sec x)(sin x cos x) (B) tan x 1 (sec x)(sin x cos x) Solution (A) Graph each side of the equation in the same viewing window (Fig. 1). FIGURE 1 4 2 2 4 Not an identity, since the graphs do not match. Try x 0 Left side: tan 0 1 1 Right side: (sec 0)(sin 0 cos 0) 1 Finding one value of x for which both sides are defined, but are not equal, is enough to verify that the equation is not an identity. (B) Graph each side of the equation in the same viewing window (Fig. 2). FIGURE 2 4 2 2 4 458 6 Trigonometric Identities and Conditional Equations The equation appears to be an identity, which we now verify: (sec x)(sin x cos x) cos1 x (sin x cos x) sin x cos x cos x cos x tan x 1 Matched Problem 7 Use a graphing utility to test whether each of the following is an identity. If an equation appears to be an identity, verify it. If the equation does not appear to be an identity, find a value of x for which both sides are defined but are not equal. (A) sin x csc x 1 cos2 x (B) sin x sec x 1 cos2 x Answers to Matched Problems In the following identity verifications, other correct sequences of steps are possible—the process is not unique. cos x cos x 1. sin x cot x sin x sin x 1 1 2. csc (x) csc x sin (x) sin x 2 sin x sin2 x cos2 x 1 3. tan x sin x cos x cos x sec x cos x cos x cos x 2 2 1 cos x sin x (1 cos x) sin x 1 2 cos x cos2 x sin2 x 4. sin x 1 cos x sin x (1 cos x) sin x (1 cos x) 2(1 cos x) 2 csc x sin x (1 cos x) 5. sec4 x 2 sec2 x tan2 x tan4 x (sec2 x tan2 x)2 12 1 cos x sin x cos2 x sin2 x cos2 x (1 cos2 x) 2 cos2 x 1 6. cot x tan x sin x cos x sin x cos x sin x cos x sin x cos x 7. (A) An identity: sin x sin x 1 csc x 1 cos2 x sin2 x sin x 4 2 2 4 (B) Not an identity; left side is not equal to the right side for x 1, for example. 4 2 2 4 6-1 EXERCISE Basic Identities and Their Use 459 6-1 A B Verify that Problems 1–24 are identities. In Problems 29–36, is the equation an identity? Explain. 1. sin sec tan 2. cos csc cot 3. cot u sec u sin u 1 4. tan csc cos 1 31. sin (x) tan x 5. cos (x) 6. cot (x) tan x 1 tan cot 7. sin csc 29. cos sec 8. tan cot 9. cot u 1 (csc u)(cos u sin u) x 1 x 30. 1 x 1 1 x 1 x 34. sin2 x cos2 x 1 35. sin4 x cos4 x 1 36. sin3 x cos3 x 1 Verify that Problems 37–64 are identities. 37. 1 (sin x cos x)2 2 cos x sin x 1 cos2 y tan2 y (1 sin y)(1 sin y) 11. cos x sin x csc x sec x sin x cos x 38. 12. cos2 x sin2 x cot x tan x sin x cos x 39. cos sin cot 1 csc 13. sin2 t cos t sec t cos t 40. sin cos tan 1 sec cos x 15. sec x 1 sin2 x cos2 t sin t csc t sin t sin u 16. csc u 1 cos2 u 17. (1 cos u)(1 cos u) sin2 u 41. 44. sec2 x csc2 x sec2 x csc2 x 20. (sin x cos x)2 1 2 sin x cos x 21. (sec t 1)(sec t 1) tan2 t 45. csc cos cot tan 46. 1 sec csc sin tan 22. (csc t 1)(csc t 1) cot2 t 23. csc x cot x 1 2 24. sec u tan u 1 2 2 In Problems 25–28, graph all parts of each problem in the same viewing window in a graphing utility. 25. x (A) y sin2 x (B) y cos2 x (C) y sin2 x cos2 x 26. x (A) y sec2 x (B) y tan2 x (C) y sec2 x tan2 x 27. x cos x (A) y cot x sin x 28. x sin x (A) y cos x tan x cos2 y 1 sin y 43. tan2 x sin2 x tan2 x sin2 x 19. cos2 x sin2 x 1 2 sin2 x 2 sin2 y 1 cos y 1 cos y (1 cos y)2 42. 1 sin y 18. (1 sin t)(1 sin t) cos2 t 32. x2 2x 1 x 1 33. sin x cos x 1 10. tan u 1 (sec u)(sin u cos u) 14. x2 4 x2 x2 (B) y 1 47. ln (tan x) ln (sin x) ln (cos x) 48. ln (cot x) ln (cos x) ln (sin x) 49. ln (cot x) ln (tan x) 50. ln (csc x) ln (sin x) 51. 1 cos A sec A 1 1 cos A sec A 1 52. 1 csc y sin y 1 1 csc y sin y 1 53. sin4 w cos4 w 1 2 cos2 w 54. sin4 x 2 sin2 x cos2 x cos4 x 1 (B) y 1 55. cos2 z 3 cos z 2 2 cos z sin2 z 1 cos z 460 6 Trigonometric Identities and Conditional Equations 56. sin2 t 4 sin t 3 3 sin t cos2 t 1 sin t 79. tan u sin u sec u 1 0 tan u sin u sec u 1 57. cos3 sin3 1 sin cos cos sin 80. tan x tan y sin x cos y cos x sin y cos x cos y sin x sin y 1 tan x tan y 58. cos3 u sin3 u 1 sin u cos u cos u sin u 81. tan cot 59. (sec x tan x)2 1 sin x 1 sin x 60. (cot u csc u)2 1 cos u 1 cos u 82. tan cot tan cot cot cot tan tan cot cot 1 1 tan tan 61. csc4 x 1 2 cot2 x cot2 x 62. sec4 x 1 2 tan2 x tan2 x Problems 83–88 require the use of a graphing utility. From the graph of y1 f(x), find a simpler function of the form g(x) k AT(x), where T(x) is one of the six trigonometric functions, that has the same graph as y1 f (x). Verify the identity f(x) g(x). 63. 1 sin v cos v cos v 1 sin v 64. sin x 1 cos x 1 cos x sin x 83. f (x) 1 sin2 x sin x cos x tan x 84. f (x) cos x 1 sin x 2 cos x 2 2 sin x 85. f (x) cos2 x 1 sin x cos2 x 86. f (x) tan x sin x 1 cos x 87. f (x) sin2 x 1 cos x 2 cos2 x 1 cos x 1 cos x 3 sin x 2 sin x cos x 1 cos x 1 cos x sin x Use a graphing utility to test whether each of Problems 65–76 is an identity. If an equation appears to be an identity, verify it. If the equation does not appear to be an identity, find a value of x for which both sides are defined but are not equal. 65. sin (x) 1 cos (x) tan (x) 66. cos (x) 1 sin x cot (x) 67. 68. cos x 1 sin (x) cot (x) 69. sin x cos2 x sec x sin x 88. f (x) 70. 1 tan2 x tan2 x 1 cot2 x 71. sin x cos2 x csc x sin x Each of the equations in Problems 89–96 is an identity in certain quadrants associated with x. Indicate which quadrants. 72. tan2 x 1 tan2 x 1 cot2 x 89. 1 cos2 x sin x 73. 1 tan x sin x 2 tan x cos x 2 91. 1 cos2 x sin x cos x cos x 74. 2 sec x 1 sin x 1 sin x 93. 1 sin2 x cos x sin x 1 cos x tan (x) 90. 1 sin2 x cos x 92. 1 sin2 x cos x 94. 1 cos 2 75. 76. 1 tan x sin x 2 tan x cos x 2 cos x cos x 2 csc x sin x 1 sin x 1 C Verify that Problems 77–82 are identities. 77. 2 sin2 x 3 cos x 3 2 cos x 1 sin2 x 1 cos x 3 cos2 z 5 sin z 5 3 sin z 2 78. cos2 z 1 sin z x sin x 95. sin x tan x 1 sin2 x 96. sin x tan x 1 sin2 x In calculus, trigonometric substitutions provide an effective way to rationalize the radical forms a 2 u 2 and a 2 u 2, which in turn leads to the solution to an important class of problems. Problems 97–100 involve such transformations. [Recall: x2 x for all real numbers x.] 97. In the radical form a 2 u 2, a 0, let u a sin x, /2 x /2. Simplify, using a basic identity, and write the final form free of radicals. 6-2 0 x /2. Simplify, using a basic identity, and write the final form free of radicals. 98. In the radical form a 2 u 2, a 0, let u a cos x, 0 x . Simplify, using a basic identity, and write the final form free of radicals. 100. In the radical form a 2 u 2, a 0, let u a cot x, 0 x /2. Simplify, using a basic identity, and write the final form free of radicals. 99. In the radical form a 2 u 2, a 0, let u a tan x, SECTION 6-2 461 Sum, Difference, and Cofunction Identities Sum, Difference, and Cofunction Identities • • • • Sum and Difference Identities for Cosine Cofunction Identities Sum and Difference Identities for Sine and Tangent Summary and Use The basic identities discussed in Section 6-1 involved only one variable. In this section we consider identities that involve two variables. • Sum and Difference Identities for Cosine We start with the important difference identity for cosine: cos (x y) cos x cos y sin x sin y (1) Many other useful identities can be readily verified from this particular one. Here, we sketch a proof of equation (1) assuming x and y are in the interval (0, 2) and x y 0. It then follows easily, by periodicity and basic identities, that (1) holds for all real numbers and angles in radian or degree measure. First, associate x and y with arcs and angles on the unit circle as indicated in Figure 1(a). Using the definitions of the circular functions given in Section 5-2, label the terminal points of x and y as shown in Figure 1(a). FIGURE 1 Difference identity. a b A(cos y, sin y) x y xy 1 1 O xy e f C (cos (x y), sin (x y)) O D(1, 0) 1 D(1, 0) 1 c d B (cos x, sin x) (a) (b) Now if you rotate the triangle AOB clockwise about the origin until the terminal point A coincides with D(1, 0), then terminal point B will be at C, as shown in Figure 1(b). Thus, since rotation preserves lengths, d(A, B) d(C, D) (c a)2 (d b)2 (1 e)2 (0 f )2 (c a)2 (d b)2 (1 e)2 f 2