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Which of the following is the Product-Sum Identity for the $\cos{\alpha} \sin{\beta}$?

A

$\cfrac{1}{2}[\sin({\alpha} + {\beta}) + \sin({\alpha} - {\beta})]$

B

$\cfrac{1}{2}[\sin({\alpha} + {\beta}) - \sin({\alpha} - {\beta})]$

C

$\cfrac{1}{2}[\sin({\alpha} + {\beta}) + \cos({\alpha} - {\beta})]$

D

$\cfrac{1}{2}[\cos({\alpha} + {\beta}) + \sin({\alpha} - {\beta})]$

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