714 CHAPTER 7 Trigonometric Identities and Equations EXAMPLE 3 Simplifying Expressions Using Double-Angle Identities Simplify each expression. (a) cos2 7x - sin2 7x (b) sin 15° cos 15° SOLUTION (a) This expression suggests one of the double-angle identities for cosine: cos 2 A = cos2 A - sin2 A. Substitute 7x for A. cos2 7x - sin2 7x = cos 217x2 = cos 14x (b) If the expression sin 15° cos 15° were 2 sin 15° cos 15°, we could apply the identity for sin 2A because sin 2 A = 2 sin A cos A. sin 15° cos 15° = 1 2 122 sin 15° cos 15° Multiply by 1 in the form 1 2 122. = 1 2 12 sin 15° cos 15°2 Associative property = 1 2 sin12 # 15°2 2 sin A cos A = sin 2 A, with A = 15° = 1 2 sin 30° Multiply. = 1 2 # 1 2 sin 30° = 1 2 = 1 4 Multiply. S Now Try Exercises 21 and 23. Identities involving larger multiples of the variable can be derived by repeated use of the double-angle identities and other identities. This is not an obvious way to begin, but it is indeed valid. EXAMPLE 4 Deriving a Multiple-Angle Identity Write sin 3x in terms of sin x. SOLUTION sin 3x = sin12x + x2 3x = 2x + x = sin 2x cos x + cos 2x sin x Sine sum identity = 12 sin x cos x2cos x + 1cos2 x - sin2 x2sin x Double-angle identities = 2 sin x cos2 x + cos2 x sin x - sin3 x Multiply. = 2 sin x11 - sin2 x2 + 11 - sin2 x2sin x - sin3 x cos2 x = 1 - sin2 x = 2 sin x - 2 sin3 x + sin x - sin3 x - sin3 x Distributive property = 3 sin x - 4 sin3 x Combine like terms. S Now Try Exercise 33. Use the simple fact that 3 = 2 + 1 here.
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