Medal and Fan, if someone really good at pre calc could help me :)

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Medal and Fan, if someone really good at pre calc could help me :)

Mathematics
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derive this identity from the sum and difference formulas for cosine: sin a sin b= (1/2){cos(a-b)-cos(a+n)}
Alright here's the thing i got 69/100 on this practice test and my teacher is letting me redo for half a credit per correct answer. can someone help explain this one to me please :)
@iGreen could you please take a look at this

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@Loser66 i can show you my work if that helps?
Go ahead
alright it might take a second there's a table and in the left hand side there's calculations, and on the right side is the reasoning
cos(a-b)-cos(a+b) over 2 reason: To manipulate the right side
yup
ok then -2 sin (-a+b)+a+b over 2) ^2 (a+a+b-b over 2)
nope
-cos s+ cost= dsin (stt over 2) sin (s-t over 2)
= sin a sin b ^2 reason : simplified
\(cos (a-b) = cos a cos b + sina sinb\\cos(a+b) = cosa cos b- sinasinb\\----------------------\\\dfrac{cos(a-b) - cos(a+b)}{2} =\dfrac{ cosacosb+sinasinb-cosacosb +sinasinb}{2}\\=\dfrac{\cancel {2} sinasinb}{\cancel 2}= sinasinb ~~=\text{left hand side}\)
So thats for the second part that i had gotten wrong right?
the first part was correct so i should keep that right? what about the reasonings?
I really appreciate your help by the way :)
\[cos (a-b) = cos a cos b + sina sinb ~~~~~(\text{identity})\\cos(a+b) = cosa cos b- sinasinb~~~(\text{identity})\\----------------------\\\dfrac{cos(a-b) - cos(a+b)}{2} \\=\dfrac{ cosacosb+sinasinb-cosacosb +sinasinb}{2}~~(\text{open parentheses}) \\=\dfrac{\cancel {2} sinasinb}{\cancel 2}= sinasinb ~~=\text{left hand side}\]
Thank you so much @Loser66, you're such a legend

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