Product-to-sum formulas: Visual walkthrough — Maths deep dive

We build from absolute zero. The only tool we are allowed to assume is the angle addition formulas, and even those we will re-picture before using.

Step 0 — What are these symbols even?

Before a single formula, let us agree what the letters mean, anchored to a picture.

WHAT. Picture a point marching anticlockwise around a circle of radius $1$ (a "unit circle"). Its position after turning through an angle is described by two numbers:

$cos (angle)$ = the point's horizontal shadow (how far right/left).
$sin (angle)$ = the point's vertical shadow (how far up/down).

WHY these. Every wave in the universe — sound, light, water — is one of these shadows drawn against time. So "multiplying two cosines" really means "multiplying two horizontal shadows". That is the object we want to understand.

PICTURE. Look at the figure: the red dot sits on the circle. Its horizontal drop-line is $cos$ , its vertical drop-line is $sin$ .

Step 1 — The one tool we're allowed: angle addition

WHAT. We start from four facts — the angle addition formulas — that tell us the shadows of a combined angle $A + B$ or $A - B$ in terms of the shadows of the two separate angles $A$ and $B$ :

$\cos(A+B)=\underbrace{\cos A\cos B}_{\text{product of horizontals}}-\underbrace{\sin A\sin B}_{\text{product of verticals}} \tag{1}$ $\cos(A-B)=\underbrace{\cos A\cos B}_{\text{same}}+\underbrace{\sin A\sin B}_{\text{sign flipped!}} \tag{2}$ $\sin(A+B)=\underbrace{\sin A\cos B}_{\text{mixed}}+\underbrace{\cos A\sin B}_{\text{mixed}} \tag{3}$ $\sin(A-B)=\underbrace{\sin A\cos B}_{\text{same}}-\underbrace{\cos A\sin B}_{\text{sign flipped!}} \tag{4}$

WHY these and nothing else. Notice the products $cos A cos B$ , $sin A sin B$ , $sin A cos B$ are already sitting inside these formulas. We don't need to invent anything — we just need to rearrange to pull a product out onto its own. That is the whole game.

PICTURE. The figure shows the single most important feature: going from $A + B$ (top) to $A - B$ (bottom), the cosine product stays put, but every sine-related term flips sign. That sign-flip is the lever we will pull.

Step 2 — Cancel by ADDING: get $cos A cos B$

WHAT. Stack equation (1) directly on top of (2) and add them, column by column.

+ cos (A + B) cos (A - B) cos (A + B) + cos (A - B) = cos A cos B - sin A sin B = cos A cos B + sin A sin B = 2 cos A cos B + 0

WHY add? Look at the sine column: $- sin A sin B$ on top, $+ sin A sin B$ below. Adding makes them destroy each other. The cosine column has the same sign twice, so it doubles. We deliberately chose the operation (addition) that kills the term we don't want.

PICTURE. Think of the two sine terms as arrows of equal length pointing opposite ways; laid tail-to-head they land back at the start — net zero. The two cosine arrows point the same way, so they stack to double length.

Now divide both sides by $2$ (to strip a lone product out of the "double"):

Step 3 — Cancel the OTHER way: get $sin A sin B$

WHAT. Same two cosine equations, but now subtract — do $(2) - (1)$ :

- cos (A - B) cos (A + B) cos (A - B) - cos (A + B) = cos A cos B + sin A sin B = cos A cos B - sin A sin B = 0 + 2 sin A sin B

WHY subtract now? This time we want to keep the sine product and kill the cosine product. The cosine column is identical in both rows, so subtracting annihilates it; the sine column, having opposite signs, reinforces into $2 sin A sin B$ . Same trick, opposite target — so opposite operation.

PICTURE. Now the cosine arrows point the same way and cancel under subtraction, while the sine arrows double. It is the mirror image of Step 2.

Divide by $2$ :

Step 4 — Mixing kinds: get $sin A cos B$

WHAT. Switch to the sine addition formulas (3) and (4) and add them:

+ sin (A + B) sin (A - B) sin (A + B) + sin (A - B) = sin A cos B + cos A sin B = sin A cos B - cos A sin B = 2 sin A cos B + 0

WHY. We want the mixed product $sin A cos B$ . It appears with a $+$ in both rows, so adding doubles it; the other mixed term $cos A sin B$ has opposite signs and cancels. The output must be a sine, because we added two sines together.

PICTURE. The $cos A sin B$ arrows point opposite → cancel; the $sin A cos B$ arrows point together → double. Note the output boxes are now sine waves, not cosine.

Divide by $2$ :

Step 5 — The fourth one: get $cos A sin B$ by subtracting

WHAT. For $cos A sin B$ , take the same two sine formulas (3) and (4), but now subtract — do $(3) - (4)$ , column by column:

- sin (A + B) sin (A - B) sin (A + B) - sin (A - B) = sin A cos B + cos A sin B = sin A cos B - cos A sin B = 0 + 2 cos A sin B

WHY subtract? Now $cos A sin B$ is the term with opposite signs across (3) and (4), so subtraction reinforces it into $2 cos A sin B$ ; the shared $sin A cos B$ column cancels. It is Step 4 run with the opposite operation.

PICTURE. Exactly the mirror of Step 4: the $sin A cos B$ arrows point the same way and cancel under subtraction, while the $cos A sin B$ arrows reinforce and double.

Divide by $2$ :

Step 6 — Every formula, checked at the corners

A formula you can't trust at the boundaries is useless. Let us check all four at their degenerate points.

Case $A = B$ (the two angles collapse into one). Put $B = A$ into Result 1: $cos A cos A = \frac{1}{2} [cos (0) + cos (2 A)] \Rightarrow cos^{2} A = \frac{1}{2} [1 + cos 2 A]$ That is the double-angle / power-reduction formula — it drops out for free! Similarly Result 2 gives $sin^{2} A = \frac{1}{2} [1 - cos 2 A]$ .

Case $B = 0$ on Result 3 (mixed, plus). $sin A cos 0 = \frac{1}{2} [sin A + sin A] = sin A ✓$ since $cos 0 = 1$ . The formula correctly gives back $sin A$ .

Case $B = 0$ on Result 4 (mixed, minus). $cos A sin 0 = \frac{1}{2} [sin A - sin A] = 0 ✓$ since $sin 0 = 0$ . The left side is $cos A \cdot 0 = 0$ too — both sides vanish, so Result 4 survives its corner just like the others.

Case $A + B = 9 0^{\circ}$ (sum angle is a right angle). Here the $^{\circ}$ flags degrees. Then $cos (A + B) = cos 9 0^{\circ} = 0$ , so one whole term switches off. In Worked Example 3 of the parent, $7 5^{\circ} + 1 5^{\circ} = 9 0^{\circ}$ is exactly why the answer came out clean as $\frac{1}{4}$ .

PICTURE. The figure shows the $A = B$ collapse: the two circle points merge into one, and the "difference angle" $A - B$ shrinks to $0$ (a $cos 0 = 1$ , the flat rightward radius), while the "sum angle" becomes $2 A$ .

The one-picture summary

Everything above is a single sentence with two knobs:

Write the two addition formulas for a kind (cos or sin), then add to keep the double-sign term or subtract to keep the opposite-sign term; divide by $2$ .

Recall Feynman: the whole walkthrough in plain words

We had four ready-made facts about the shadows of two combined angles $A$ and $B$ . Each fact is a little sum of two products. We noticed that between the " $A + B$ " fact and the " $A - B$ " fact, one product keeps its sign and the other flips. So: if we add the two facts, the flipping product cancels to nothing and the steady one doubles. If we subtract, the reverse happens. Either way, one lonely product is left standing — and we halve it to set it free. Do this with the two cosine facts and you get the cos·cos and sin·sin rules (both spit out cosines). Do it with the two sine facts and you get the two mixed rules (both spit out sines). Check the corners — same angle, zero angle, right-angle sum — and the machine never jams; it even hands you the double-angle formulas as a bonus.

Recall

Which operation on the two cosine addition formulas gives

cos A cos B

Adding them — the

sin A sin B

terms cancel, the

cos A cos B

doubles.

Which operation gives

sin A sin B

Subtracting,

cos (A - B) - cos (A + B)

— the cosine product cancels.

Which operation on the two sine formulas gives

cos A sin B

Subtracting,

sin (A + B) - sin (A - B)

— the

sin A cos B

terms cancel.

Why does subtracting flip a sign?

Replacing

B

- B

mirrors a point across the horizontal axis, negating its vertical shadow

sin B

What identity appears when you set

A = B

in the

cos A cos B

rule?

cos^{2} A = \frac{1}{2} [1 + cos 2 A]

, the power-reduction / double-angle formula.

What are

A

and

B

, and what units do we use by default?

Two ordinary angles (each a turn on the unit circle); radians unless a

^{\circ}

mark says degrees.

Same-kind inputs (cos·cos, sin·sin) produce sums of what?

Cosines.

Mixed inputs (sin·cos, cos·sin) produce sums of what?

Sines.

Why divide by

2

at the end of every derivation?

Adding/subtracting two addition formulas gives

2 \times

the product, so halving isolates it.

Connections

Product-to-sum formulas — the parent note this page zooms into.
Angle Addition Formulas — the single tool every step rests on.
Sum-to-Product Formulas — the exact inverse, derived by the same add/subtract trick.
Double Angle Formulas — falls out of Step 6 when $A = B$ .
Integration of Trig Functions — why we bother turning products into sums.
Wave Interference and Beats — the physical picture of un-mixing frequencies.
Fourier Series — orthogonality proofs use exactly these identities.