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We are asked to simplify the expression: sin x + sin 3x + sin 5x using product-to-sum identities . We first group the terms as follows: (sin x + sin 5x) + sin 3x Now apply the identity: sin A + sin B = 2 sin[(A + B)/2] cos[(A − B)/2] Apply this to sin x + sin 5x: A = 5x, B = x sin x + sin 5x = 2 · sin[(5x + x)/2] · cos[(5x − x)/2] = 2 · sin(3x) · cos(2x) Now substitute this back into the expression: sin x + sin 3x + sin 5x = [2 · sin(3x) · cos(2x)] + sin 3x Now factor sin 3x from both terms: = sin 3x · [2 cos 2x + 1]
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