Sum-to-product formulas: Visual walkthrough — Maths deep dive

Before we start, one word we will use constantly:

Step 1 — Two angles, and the two combinations that matter

WHAT. Pick two angles. Instead of calling them $A$ and $B$ right away, call them $x + y$ and $x - y$ . That is not a trick yet — it is just a naming choice that will pay off.

WHY. The angle addition formulas are built for exactly this shape: they tell us what happens to $sin$ and $cos$ of $x + y$ and of $x - y$ . If we feed the machine those two inputs, magic terms will cancel in Step 2.

PICTURE. Look at the figure. The blue angle is $x$ . We tilt up by $y$ to reach $x + y$ (yellow), and down by $y$ to reach $x - y$ (green). So $y$ is the half-spread between the two angles, and $x$ sits exactly in the middle.

$x$ — the middle angle (the average of our two angles).
$y$ — the half-spread: how far each angle leans off the middle.
$x + y$ — the upper angle. · $x - y$ — the lower angle.

Step 2 — Add the two sine formulas (watch a term vanish)

WHAT. Write both sine-addition formulas and stack them:

$sin (x + y) = term P sin x cos y + term Q cos x sin y$ $sin (x - y) = term P sin x cos y - term Q cos x sin y$

Here each symbol is:

$sin x cos y$ (term P) — appears with a $+$ in both rows.
$cos x sin y$ (term Q) — appears $+$ in the top row, $-$ in the bottom row.

Adding the rows, the two Q's are $+ Q$ and $- Q$ : they cancel. The two P's survive and double:

$\sin(x+y) + \sin(x-y) = 2\sin x\cos y \tag{1}$

WHY. We added precisely because the odd (sign-flipping) term is the one we want gone. Addition is a filter: it keeps the part that is the same in both rows, kills the part that flips.

PICTURE. The figure shows the two expansions as two horizontal bars of coloured blocks. Term Q is red and points opposite ways in the two bars — stacking them, the red arrows annihilate. Term P is blue and points the same way — it stacks to double height.

Step 3 — Subtract instead, to catch the other term

WHAT. Same two rows, but now subtract the bottom from the top:

$\sin(x+y) - \sin(x-y) = 2\cos x\sin y \tag{2}$

Term by term: the two $sin x cos y$ (P) blocks are identical, so subtracting kills them; the $cos x sin y$ (Q) blocks are $+ Q$ and $- (- Q) = + Q$ , so they double.

WHY. Subtraction is the complementary filter to Step 2: it keeps what flips and kills what stays. Between add (Step 2) and subtract (Step 3) we have now captured both products $sin x cos y$ and $cos x sin y$ . Nothing about these two angles is left unexplored.

PICTURE. Mirror of the last figure: this time the blue P-blocks cancel and the red Q-blocks survive and double.

Step 4 — Do the same for cosine (all four cases now covered)

WHAT. The cosine-addition formulas differ by one sign, so their sum and difference behave differently:

$cos (x + y) = cos x cos y - sin x sin y$ $cos (x - y) = cos x cos y + sin x sin y$

$cos x cos y$ — enters $+$ in both rows.
$sin x sin y$ — enters $-$ on top, $+$ on bottom (note: cosine's cross term is negative up top).

Add ⇒ the $sin x sin y$ terms ( $- S$ and $+ S$ ) cancel: $\cos(x+y)+\cos(x-y) = 2\cos x\cos y \tag{3}$

Subtract ( $cos (x - y) - cos (x + y)$ , chosen so the surviving term is positive) ⇒ the $cos x cos y$ terms cancel: $\cos(x-y)-\cos(x+y) = 2\sin x\sin y \tag{4}$

WHY. Cosine's addition rule carries a minus on its cross term. That minus is the entire reason $cos A - cos B$ will end up with a leading minus sign later — it is born right here, not invented at the end.

PICTURE. Two mini-panels: the top panel adds (green $sin x sin y$ blocks annihilate); the bottom subtracts (yellow $cos x cos y$ blocks annihilate). The red minus badge on the $sin sin$ block is highlighted — remember where it lives.

Step 5 — The rename that finishes the job

WHAT. Equations (1)–(4) still speak in $x + y$ and $x - y$ . We now rename: $A = x + y, B = x - y .$ Solve this little 2-equation system:

Add them: $A + B = 2 x \Rightarrow x = \frac{A + B}{2}$ — the half-sum.
Subtract them: $A - B = 2 y \Rightarrow y = \frac{A - B}{2}$ — the half-difference.

WHY. The left side of (1) is $sin (x + y) + sin (x - y)$ . After renaming that is literally $sin A + sin B$ — a plain sum of the two functions we care about! The rename converts "sum of functions" into "product with half-sum and half-difference arguments." That is the payoff promised in Step 1: choosing $x$ = middle, $y$ = half-spread makes $x = \frac{A + B}{2}$ and $y = \frac{A - B}{2}$ automatically.

PICTURE. A number line: $A$ and $B$ marked; their midpoint $\frac{A + B}{2}$ (blue) sits dead centre, and the arrow of length $\frac{A - B}{2}$ (yellow) reaches from the midpoint out to $A$ . This is the same $x$ -in-the-middle, $y$ -half-spread picture from Step 1, now labelled in $A, B$ .

$\frac{A + B}{2}$ — the centre of the two angles (blue dot).
$\frac{A - B}{2}$ — half the gap between them (yellow arrow).

Step 6 — Substitute and read off all four identities

WHAT. Replace $x \to \frac{A + B}{2}$ and $y \to \frac{A - B}{2}$ , and $sin (x \pm y) \to sin A, sin B$ (etc.) in (1)–(4):

Term-by-term for the first one: the leading $2$ came from the doubling in Step 2; $sin \frac{A + B}{2}$ is the old $sin x$ ; $cos \frac{A - B}{2}$ is the old $cos y$ .

WHY. Each identity is just one of (1)–(4) wearing new clothes. The minus in the fourth line is the same minus that survived Step 4 — geometry, not magic.

PICTURE. A summary table-figure pairs each equation with a tiny icon of which term cancelled to make it.

Step 7 — Degenerate case: when $A = B$

WHAT. What if the two angles are equal? Set $A = B$ . Then $\frac{A + B}{2} = A$ and $\frac{A - B}{2} = 0$ .

$sin A + sin A = 2 sin A cos 0 = 2 sin A (cos 0 = 1) ✓$ $cos A - cos A = - 2 sin A sin 0 = 0 (sin 0 = 0) ✓$

WHY. A formula you cannot trust at its boundaries is a formula you cannot trust. Here $A = B$ gives the obvious answers ( $sin A + sin A = 2 sin A$ ; a thing minus itself is $0$ ), so the identity survives its own edge case. This limit is also the doorway to the Double Angle Formulas — feed $A = B$ into the cosine-difference-of-arguments route and doubling appears.

PICTURE. The half-spread arrow $\frac{A - B}{2}$ shrinks to a point as $B$ slides onto $A$ ; $cos$ of it climbs to $1$ , $sin$ of it falls to $0$ .

The one-picture summary

WHAT. One figure carries the entire journey: add/subtract the addition formulas → a term cancels → rename to half-sum & half-difference → read off the product.

Recall Feynman: tell the whole story in plain words

I have two angles. I decide to think of them not as "this one and that one," but as a middle and a spread: $x$ is where they balance, $y$ is how far each leans off. Now I use the angle-addition rules on the upper angle $x + y$ and the lower angle $x - y$ . Each rule splits into two pieces. When I add the upper and lower results, one piece appears with opposite signs and dies; the surviving piece doubles. When I subtract, the other piece dies and the first doubles. So from four add/subtract combos I get four clean " $2 \times$ something $\times$ something" statements. Finally I rename: the middle $x$ was always $\frac{A + B}{2}$ and the spread $y$ was always $\frac{A - B}{2}$ . Swap those in, and the left side turns into the plain sum $sin A + sin B$ I actually wanted. The right side is now a product — and a product is a machine for solving equations, because if it equals zero, one of its factors must be zero. The one weird sign — the minus in $cos A - cos B$ — isn't a special rule to memorize; it's just cosine's own minus sign from Step 4, riding along to the end.

Recall Quick self-test

Why does adding the sine formulas kill $cos x sin y$ ? ::: Because that term is $+$ on top and $-$ on the bottom, so the two cancel; only $sin x cos y$ survives and doubles. Where does the minus in $cos A - cos B$ come from? ::: From cosine's negative cross term $- sin x sin y$ in the addition formula, surviving through the subtraction in Step 4. What are $x$ and $y$ in terms of $A, B$ ? ::: $x = \frac{A + B}{2}$ (half-sum), $y = \frac{A - B}{2}$ (half-difference). Check $sin A + sin B$ at $A = B$ . ::: $2 sin A cos 0 = 2 sin A$ , matching $sin A + sin A$ .

Connections

Angle Addition Formulas — the single seed everything grew from.
Product-to-Sum Formulas — run this whole picture backwards.
Double Angle Formulas — the $A = B$ limit of Step 7.
Solving Trigonometric Equations — why we wanted a product at all.
Beats and Superposition (Waves) — the physical face of the product form.