Question:
sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin
Answer:
Given sin x + sin 3x + sin 5x + sin 7x
= 2*sin(x+3x)/2 * cos(x-3x)/2 + 2*sin(5x+7x)/2 * cos(5x-7x)/2
= 2*sin(4x/2) * cos(-2x/2) + 2*sin(12x/2) * cos(-2x/2)
= 2*sin2x * cos(-x) + 2*sin6x * cos(-x)
= 2*sin2x * cosx + 2*sin6x * cosx
= 2*cosx *(sin2x + sin6x)
= 2*cosx *{sin(2x+6x)/2 * cos(2x-6x)/2}
= 2*cosx *{2*sin(8x/2) * cos(-4x/2)}
= 2*cosx *{2*sin4x * cos(-2x)}
= 4*cosx *sin4x * cos2x
= 4*cosx *cos2x*sin4x
So sin x + sin 3x + sin 5x + sin 7x = 4*cosx *cos2x*sin4x
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