684 CHAPTER 7 Trigonometric Identities and Equations CAUTION When taking the square root, be sure to choose the sign based on the quadrant of U and the function being evaluated. EXAMPLE 2 Writing OneTrigonometric Function inTerms of Another Write cos x in terms of tan x. SOLUTION By identities, sec x is related to both cos x and tan x. 1 + tan2 x = sec2 x Pythagorean identity 1 1 + tan2 x = 1 sec2 x Take reciprocals. 1 1 + tan2 x = cos2 x The reciprocal of sec2 x is cos2 x. {B 1 1 + tan2 x = cos x Take the square root of each side. cos x = {1 2 1 + tan2 x Quotient rule for radicals: 3n a b = 1na 2n b ; Rewrite. cos x = {21 + tan2 x 1 + tan2 x Rationalize the denominator. The choice of the + sign or the - sign depends on the quadrant of x. S Now Try Exercise 47. Remember both the positive and negative roots. Figure 2 supports the identity sin2 x + cos2 x = 1. ■ y1 = sin 2 x + cos2 x y2 = 1 −4 4 11p 4 11p 4 − Figure 2 With an identity, there should be no difference between the two graphs. (b) tan u = sin u cos u c os u tan u = sin u Multiply each side by cos u. a 1 sec ub tan u = sin u Reciprocal identity ¢- 3234 34 ≤ a- 5 3b = sin u sin u = 5234 34 Multiply and rewrite. (c) cot1-u2 = 1 tan1-u2 cot1-u2 = 1 -tan u Even-odd identity cot1-u2 = 1 -A - 5 3B tan u = - 5 3 cot1-u2 = 3 5 1 -A - 5 3B = 1 , 5 3 = 1 # 3 5 = 3 5 S Now Try Exercises 11, 19, and 31. tan u = - 5 3 , and from part (a), 1 sec u = 1 - 234 3 = - 32 34 = - 32 34 # 234 234 = - 3 234 34 . Quotient identity that relates the tangent and sine functions Reciprocal identity that relates the tangent and cotangent functions
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