Formulae for Conversion of Product into Sum or Difference

Conversion of product into sum or difference

Theorem: for any angles $A$ and $B$,

These are exactly the four identities used to derive the sum-to-product formulas in section 3.4.1, read in the

reverse direction: instead of starting from $\sin(A+B)\pm\sin(A-B)$ and simplifying to a product, we start from

a product $2\sin A\cos B$ (etc.) and expand it into the sum/difference $\sin(A+B)+\sin(A-B)$ using the ordinary

compound-angle formulas of section 3.1, added or subtracted in pairs.

Solved Examples

Ex.2(i) — Express $2\sin4\theta\cos2\theta$ as a sum or difference. Direct substitution into

$2\sin A\cos B=\sin(A+B)+\sin(A-B)$ with $A=4\theta,B=2\theta$: $=\sin6\theta+\sin2\theta$.

Ex.2(ii) — Express $4\sin\dfrac{A+B}{2}\sin\dfrac{A-B}{2}$ as a sum or difference. This is $2\times\left(2\sin \dfrac{A+B}2\sin\dfrac{A-B}2\right)$; apply $2\sin X\sin Y=\cos(X-Y)-\cos(X+Y)$ with $X=\dfrac{A+B}2,Y=\dfrac{A-B}2$:

$X-Y=B,\ X+Y=A$, so the product is $2[\cos B-\cos A]=2\cos B-2\cos A$. …