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 #
