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Derive this identity from the sum and difference formulas for cosine: sin a sin b = (1...

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atrinity | (Level 1) Honors

Posted April 20, 2013 at 11:21 PM via web

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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 + b)]

Start with the right-hand side since it is more complex.


Please provide a reason with each step

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violy | High School Teacher | (Level 3) Assistant Educator

Posted April 21, 2013 at 12:40 AM (Answer #1)

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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]`

Multiply. 

`1/2[2sinasinb] = sinasinb`

Hence, right hand side is equal to left hand side. 

 

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oldnick | (Level 1) Valedictorian

Posted April 21, 2013 at 1:03 AM (Answer #2)

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Prostapheresis formula shows that:

`cos(a-b)-cos(a+b)=-2sin((a-b+a+b)/2)sin(((a-b)-(a+b))/2)`

`=-2sin a sin(-b)=2sina sin b`

`thus:`

`1/2(cos(a-b)-cos(a+b))=sin a sin b`

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