Question #01a2a

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Answer 1 · 2018-01-30T07:58:54

#1/2(cos(5theta)-cos(9theta))#

Using the sum and difference formulas for cosine we have:

#cos(x-y)=cos(x)cos(y)+sin(x)sin(y)#
#cos(x+y)=cos(x)cos(y)-sin(x)sin(y)#

If we subtract those formulas:

#cos(x-y)-cos(x+y) =2sin(x)sin(y)#

so

#sin(x)sin(y) = 1/2(cos(x-y)-cos(x+y))#

so

#sin(2theta)sin(7theta) = 1/2(cos(2theta-7theta)-cos(2theta+7theta))#

#=1/2(cos(-5theta)-cos(9theta))#

since cosine is even we can rewrite this as:

#=1/2(cos(5theta)-cos(9theta))#