716 CHAPTER 7 Trigonometric Identities and Equations EXAMPLE 6 Using a Product-to-Sum Identity Write 4 cos 75° sin 25° as the sum or difference of two functions. SOLUTION 4 cos 75° sin 25° = 4 c 1 2 1sin175° + 25°2 - sin175° - 25°22d = 2 sin 100° - 2 sin 50° Simplify. S Now Try Exercise 37. Use the identity for cos A sin B, with A = 75° and B = 25°. We can transform the product-to-sum identities into equivalent useful forms — the sum-to-product identities — using substitution. Consider the product-to-sum identity for sin A cos B. sin A cos B = 1 2 3sin 1A + B2 + sin 1A - B24 Let u = A + B and v = A - B. Then u + v = 2A and u - v = 2B, so A = u + v 2 and B = u - v 2 . Substituting for A and B in the above product-to-sum identity gives the following. sin a u + v 2 b cos a u - v 2 b = 1 2 1sin u + sin v2 Substitute. sin u + sin v = 2 sin a u + v 2 b cos a u - v 2 b The other three sum-to-product identities are derived using the same substitutions into the other three product-to-sum formulas. Product-to-sum identity Use substitution variables to write the product-to-sum identity in terms of u and v. Multiply by 2. Interchange sides. Sum-to-Product Identities sin A +sin B =2 sin a A +B 2 b cos a A −B 2 b sin A −sin B =2 cos a A +B 2 b sin a A −B 2 b cos A +cos B =2 cos a A +B 2 b cos a A −B 2 b cos A −cos B =−2 sin a A +B 2 b sin a A −B 2 b
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