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When deriving the Sum to Product Identities for Sine and Cosine of (a + b) (a - b) we take X= (a + b) and Y = (a - b) and then ÷ by 2 when adding X + Y and X - Y. Is this because we have two values of a and b? When we are given say Sine of 105 degrees and 15 degrees we just add them and then take the difference rather than (a + b) + (a - b) and (a + b) - (a - b). We still divide by 2. Why is this?

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    $\begingroup$ If you just write it out, you can see that $(X+Y)/2 = a$ and $(X-Y)/2 = b$. If that doesn't answer your question, can you add more context? I'm not sure what you're really asking. $\endgroup$ – Matthew Leingang Mar 10 '17 at 13:37
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It's because $\begin{align*}a-b&=x\\ a+b&=y\end{align*}$ so adding the two equations yields $2a=x+y\to a=\frac{x+y}{2}$ the same applies with $y$

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  • $\begingroup$ Thanks. I saw a video on You Tube and I posted my question which stated what you said. Most of the videos tell the students to make the substitution but give no explanation why. Thanks for your response $\endgroup$ – Shaun Kirk Apr 12 '17 at 9:09
  • $\begingroup$ Thanks. I saw a video on You Tube after I posted my question which stated what you said. Most of the videos tell the students to make the substitution but give no explanation why. Thank for your response $\endgroup$ – Shaun Kirk Apr 12 '17 at 9:11

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