508 CHAPTER 7 Analytic Trigonometry CAUTION Do not try to establish an identity by treating it as an equation. We cannot add or multiply both sides by the same expression because we do not know if the sides are equal. That is what we are trying to prove. j Guidelines for Establishing Identities • It is almost always preferable to start with the side containing the more complicated expression. • Rewrite sums or differences of quotients as a single quotient. • Sometimes it helps to rewrite one side in terms of sine and cosine functions only. • Always keep the goal in mind. As you manipulate one side of the expression, keep in mind the form of the expression on the other side. ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 7.4 Assess Your Understanding 1. True or False sin 1 cos . 2 2 θ θ = − (pp. 417–419) 2. True or False sin cos cos sin . θ θ θ θ ( ) ( ) − + − = − (pp. 422–423) Establishing an Identity Establish the identity: 1 sin cos cos 1 sin θ θ θ θ − = + Solution EXAMPLE 8 Start with the left side and multiply the numerator and the denominator by 1 sin .θ + (Alternatively, we could multiply the numerator and the denominator of the right side by 1 sin .θ − ) 1 sin cos 1 sin cos 1 sin 1 sin 1 sin cos 1 sin cos cos 1 sin cos 1 sin 2 2 θ θ θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) − = − ⋅ + + = − + = + = + Multiply the numerator and the denominator by θ +1 sin . θ θ − = 1 sin cos 2 2 Divide out θ cos . Now Work PROBLEM 55 Although practice is the only real way to learn how to establish identities, the following guidelines should prove helpful. Concepts and Vocabulary 3. Suppose that f and g are two functions with the same domain. If f x g x ( ) ( ) = for every x in the domain, the equation is called a(n) . Otherwise, it is called a(n) equation. 4. tan sec 2 2 θ θ − = . 5. cos cos θ θ ( ) − − = . 6. True or False sin sin 0 θ θ ( ) − + = for any value of .θ 7. True or False In establishing an identity, it is often easiest to just multiply both sides by a well-chosen nonzero expression involving the variable. 8. True or False tan cos sin θ θ θ ⋅ = for any k2 1 2 . θ π ( ) ≠ + 9. Multiple Choice Which of the following equations is not an identity? (a) cot 1 csc 2 2 θ θ + = (b) tan tan θ θ ( ) − = − (c) tan cos sin θ θ θ = (d) csc 1 sin θ θ = 10. Multiple Choice The expression 1 1 sin 1 1 sin θ θ − + + simplifies to which of the following? (a) 2cos2 θ (b) 2 sec2 θ (c) 2 sin2 θ (d) 2csc2 θ 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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