Question 1085917
<font color="black" face="times" size="3">The trig identity we'll use is 
*[Tex \Large \sin(\alpha)\cos(\beta) = \frac{1}{2}\left(\sin(\alpha+\beta)+\sin(\alpha-\beta)\right)]
which is from this <a href="http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf">list of trig identities</a> (see the "Product to Sum Formulas" section on page 2)

In this case,
*[Tex \Large \alpha] = alpha = x+6y
*[Tex \Large \beta] = beta = x-6y


So this means,
*[Tex \Large \sin(\alpha)\cos(\beta) = \frac{1}{2}\left(\sin(\alpha+\beta)+\sin(\alpha-\beta)\right)]
*[Tex \Large \sin(x+6y)\cos(x-6y) = \frac{1}{2}\left(\sin(x+6y+x-6y)+\sin(x+6y-(x-6y))\right)]
*[Tex \Large \sin(x+6y)\cos(x-6y) = \frac{1}{2}\left(\sin(2x)+\sin(x+6y-x+6y)\right)]
*[Tex \Large \sin(x+6y)\cos(x-6y) = \frac{1}{2}\left(\sin(2x)+\sin(12y)\right)]
*[Tex \Large \sin(x+6y)\cos(x-6y) = \frac{1}{2}\sin(2x)+\frac{1}{2}\sin(12y)]
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