Sum-to-Product Formulas

The four sum-to-product formulas rewrite a sum or difference of two

sine or cosine terms as a product -- the reverse operation to the sum/difference formulas of

Section 6, and essential for simplifying expressions and factoring trigonometric equations

before solving them (Section 10).

Setting up the substitution. Let $x$ and $y$ be any two real numbers, and set

Applying the Section 6 sum and difference formulas to $\sin(p+q)$ and $\sin(p-q)$:

Theorem ($\sin x+\sin y$ and $\sin x-\sin y$). Adding the two equations above, the

$\cos p\sin q$ terms cancel:

Subtracting them instead, the $\sin p\cos q$ terms cancel:

Theorem ($\cos x+\cos y$ and $\cos x-\cos y$). Applying the same substitution to cosine,

Adding, the $\sin p\sin q$ terms cancel:

Subtracting ($\cos x$ minus $\cos y$), the $\cos p\cos q$ terms cancel and a minus sign is left

over:

Note carefully the one formula with a minus sign out front, $\cos x-\cos y=-2\sin(\ldots)\sin(\ldots)$

-- this is the single most common sign slip made when applying these four formulas, since the

other three all have a plain $+2$ or $2$ in front.

Summary of all four.

$\sin x+\sin y = 2\sin\tfrac{x+y}{2}\cos\tfrac{x-y}{2}, \qquad \sin x-\sin y = 2\cos\tfrac{x+y}{2}\sin\tfrac{x-y}{2},$ …