We're getting into the sum of sines and product of sines identities, which are relatively obscure. Let's start by listing them; maybe I'll derive them at the end.
sin a + sin b = 2 sin ((a+b)/2)cos((a-b)/2)
sin a sin b = 1/2 ( cos(a-b) - cos(a+b))
Let's apply them.
sin theta ( sin 4 theta + sin 6 theta)
= sin theta ( 2 sin (( 4 theta + 6 theta)/2) cos((4 theta - 6 theta )/2))
= 2 sin theta sin(5 theta) cos theta
= 2 cos theta ( sin theta sin 5 theta)
= 2 cos theta(1/2 ( cos(theta - 5 theta) - cos (theta + 5 theta)))
= cos theta(cos 4 theta - cos 6 theta)
I gotta go; derivations later maybe.
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OK, let's derive the two identities we used, first:
sin a sin b = 1/2 ( cos(a-b) - cos(a+b))
This comes from the cosine sum and difference angle formulas:
cos(a-b)=cos a cos b + sin a sin b
cos(a+b) = cos a cos b - sin a sin b
Subtracting,
cos(a-b) - cos(a+b) = 2 sin a sin b
sin a sin b = 1/2( cos(a-b) - cos(a+b) ) quad sqrt
Next,
sin a + sin b = 2 sin ((a+b)/2)cos((a-b)/2)
This comes from the sine sum and difference angle formulas:
sin(x+y) = sin x cos y + cos x sin y
sin(x-y) = sin x cos y - cos x sin y
Adding,
sin(x+y) + sin(x-y) = 2 sin x cos y
Let a=x+y, b=x-y so a+b=2x or x=(a+b)/2 and y=(a-b)/2
sin a + sin b = 2 sin((a+b)/2) cos((a-b)/2) quad sqrt
