690 CHAPTER 7 Trigonometric Identities and Equations EXAMPLE 1 Verifying an Identity (Working with One Side) Verify that the following equation is an identity. cot u + 1 = csc u1cos u + sin u2 SOLUTION We use the fundamental identities to rewrite one side of the equation so that it is identical to the other side. The right side is more complicated, so we work with it, as suggested in Hint 2, and use Hint 3 to change all functions to expressions involving sine or cosine. Right side of given equation $11111111%11111111& csc u1cos u + sin u2 = 1 sin u 1cos u + sin u2 csc u = 1 sin u = cos u sin u + sin u sin u Distributive property: a1b + c2 = ab + ac = cot u + 1 cos u sin u = cot u; sin u sin u = 1 (11)11* Left side of given equation The right side is identical to the left side, so the given equation is an identity. S Now Try Exercise 45. −4 4 11p 4 For u = x, y1 = cot x + 1 y2 = csc x(cos x + sin x) 11p 4 − The graphs coincide, which supports the conclusion in Example 1. −4 4 11p 4 y1 = tan 2 x(1 + cot2 x) 1 1 − sin2 x y2 = 11p 4 − The screen supports the conclusion in Example 2. EXAMPLE 2 Verifying an Identity (Working with One Side) Verify that the following equation is an identity. tan2 x11 + cot2 x2 = 1 1 - sin2 x SOLUTION We work with the more complicated left side, as suggested in Hint 2, and use the fundamental identities to obtain the right side. Left side of given equation $1111111%1111111& tan2 x11 + cot2 x2 = tan2 x + tan2 x cot2 x Distributive property = tan2 x + tan2 x # 1 tan2 x cot2 x = 1 tan2 x = tan2 x + 1 tan2 x # 1 tan2 x = 1 = sec2 x Pythagorean identity = 1 cos2 x sec2 x = 1 cos2 x = 1 1 - sin2 x Pythagorean identity (11)11* Right side of given equation The left side is identical to the right side, so the given equation is an identity. S Now Try Exercise 49.
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