This is the derivation
sin (x+y)=sin x cos y+ cos x sin y" " " "1st equation
sin (x-y)=sin x cos y- cos x sin y" " " "2nd equation
subtract 2nd from the 1st
sin (x+y)-sin(x-y)=2*cos x sin y
it follows
cos x sin y=1/2[sin (x+y)-sin(x-y)]
Now, let x=pi/3 and y=(7pi)/8
cos (pi/3) sin ((7pi)/8)=1/2[sin (pi/3+(7pi)/8)-sin(pi/3-(7pi)/8)]
cos (pi/3) sin ((7pi)/8)=1/2[sin ((8pi+21pi)/24)-sin ((8pi-21pi)/24)]
cos (pi/3) sin ((7pi)/8)=1/2[sin ((29pi)/24)-sin ((-13pi)/24)]
cos (pi/3) sin ((7pi)/8)=1/2[sin (pi+(5pi)/24)-sin ((-13pi)/24)]
Note: sin ((-13pi)/24)=-sin ((13pi)/24)
so that
cos (pi/3) sin ((7pi)/8)=1/2[sin (pi+(5pi)/24)-(-sin ((13pi)/24))]
cos (pi/3) sin ((7pi)/8)=1/2[sin (pi+(5pi)/24)+sin ((13pi)/24)]
cos (pi/3) sin ((7pi)/8)=
1/2[sin pi cos((5pi)/24)+cos pi sin ((5pi)/24)+sin ((13pi)/24)]
cos (pi/3) sin ((7pi)/8)=1/2[sin((13pi)/24)-sin ((5pi)/24)]
final answer
1/2*sin((13pi)/24)-1/2*sin ((5pi)/24)
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