How to derive the sum to product formula

We shall look at an easy method of remembering or knowing how to derive the sum to product formula. As you see below is the product to sum formula:

Screen Shot 2015-07-07 at 2.43.34 pm

Credits: SEAB

We first let
\alpha = {\frac{1}{2}({P+Q})}
\beta = {\frac{1}{2}({P-Q})}, and thus have,
\alpha + \beta = P
\alpha - \beta = Q

The first formula should look like this:
sin(\alpha+\beta) + sin(\alpha-\beta)=2{\mathrm{sin}\alpha}{\mathrm{cos}\beta}

\frac{1}{2} is being taken care of too but be careful that there is a “2” there.

This little switcheroo will allow us to be at the doorsteps of the sum to product formula. Students should attempt the substitution yourself, and you will realise that it is not that hard after all. This should ease your life of finding P’s and Q’s. Try deriving the remaining three formulas!

Please comment at the bottom if you run into any questions, I will love to help! 🙂

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