Derive this identity from the sum and difference formulas for cosine:
sin a sin b = (1 / 2)[cos(a – b) – cos(a + b)]
Start with the right-hand side since it is more complex.
Please provide a reason with each step
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We use the formula for sum and difference of two angles for cosine.
`cos(a - b) = cosacosb + sinasinb`
`cos(a + b) = cosacosb - sinasinb`
So, we will have:
`1/2[cosacosb + sinasinb - (cosacosb - sinasinb)] = 1/2[cosacosb + sinasinb - cosacosb + sinasinb]`
Combine like terms.
`1/2[2sinasinb] = sinasinb`
Hence, right hand side is equal to left hand side.
Prostapheresis formula shows that:
`=-2sin a sin(-b)=2sina sin b`
`1/2(cos(a-b)-cos(a+b))=sin a sin b`
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