Sum to Product Formulas — Trig Identities, Proof

Four Formulas That Turn Stubborn Sums Into Easy Products

When two sines won't add to a closed form by inspection, factor them. The sum-to-product family is the trigonometric analogue of factoring a polynomial sum.

The sum to product formula family is a set of four trigonometric identities:

• $\sin A + \sin B$ as a product.

• $\sin A - \sin B$ as a product.

• $\cos A + \cos B$ as a product.

• $\cos A - \cos B$ as a product.

Each rewrites a sum or difference of two sinusoids as twice the product of a sine (or cosine) of the half-sum and a sine (or cosine) of the half-difference.

The Four Formulas

$$\boxed{;\sin A + \sin B = 2 \sin!\left(\dfrac{A+B}{2}\right) \cos!\left(\dfrac{A-B}{2}\right);}$$

$$\boxed{;\sin A - \sin B = 2 \cos!\left(\dfrac{A+B}{2}\right) \sin!\left(\dfrac{A-B}{2}\right);}$$

$$\boxed{;\cos A + \cos B = 2 \cos!\left(\dfrac{A+B}{2}\right) \cos!\left(\dfrac{A-B}{2}\right);}$$

$$\boxed{;\cos A - \cos B = -2 \sin!\left(\dfrac{A+B}{2}\right) \sin!\left(\dfrac{A-B}{2}\right);}$$

The pattern: the half-sum angle goes inside the function on the left of the product; the half-difference angle goes inside the function on the right. The pairing of functions follows a memorable rhyme — sin+sin gives sin·cos, sin−sin gives cos·sin, cos+cos gives cos·cos, cos−cos gives sin·sin (with a negative sign).

Quick facts.

• Hold for all real $A, B$ — no domain restrictions.

• Companion identities — product to sum: the reverse direction is $2 \sin X \cos Y = \sin(X+Y) + \sin(X-Y)$, $2 \cos X \cos Y = \cos(X+Y) + \cos(X-Y)$, $2 \sin X \sin Y = \cos(X-Y) - \cos(X+Y)$.

• Derived from sum-and-difference identities (no new tools needed).

• In radians: every angle in the formulas is in radians by default for calculus contexts; degrees work identically for school contexts. The formula structure does not change.

• Grade introduced: CCSS-M F-TF.C.9 — extending sum-and-difference identities; NCERT Class 11 Chapter 3 — Trigonometric Functions.

Double-Anchoring — Right Triangle and Unit Circle

The proof comes directly from the sum-and-difference identities, which themselves rest on the unit circle.

The proof of $\sin A + \sin B = 2 \sin!\left(\dfrac{A+B}{2}\right) \cos!\left(\dfrac{A-B}{2}\right)$.

Let $X = (A+B)/2$ and $Y = (A-B)/2$. Then $X + Y = A$ and $X - Y = B$. Apply the sine sum and difference formulas:

$$\sin(X + Y) = \sin X \cos Y + \cos X \sin Y = \sin A.$$ $$\sin(X - Y) = \sin X \cos Y - \cos X \sin Y = \sin B.$$

Add the two equations:

$$\sin A + \sin B = 2 \sin X \cos Y = 2 \sin!\left(\dfrac{A+B}{2}\right) \cos!\left(\dfrac{A-B}{2}\right).$$

Two lines.

The other three identities follow from the same substitution, subtracting the equations (for the difference identities) or pairing with cosine sum-and-difference (for the cosine identities).

On the unit circle. When $A$ and $B$ are close, $(A-B)/2$ is small, $\cos((A-B)/2) \approx 1$, and the sum $\sin A + \sin B \approx 2 \sin((A+B)/2)$ — the average of the two angles' sines, doubled. The unit-circle picture: two nearby points on the circle have an "average" point at the half-sum angle, and the chord between them shrinks like $\sin((A-B)/2)$.

Three Worked Examples — Quick, Standard, Stretch

Quick. Evaluate $\sin 75° + \sin 15°$ exactly.

Apply $\sin A + \sin B = 2 \sin((A+B)/2) \cos((A-B)/2)$ with $A = 75°$, $B = 15°$:

$$\sin 75° + \sin 15° = 2 \sin!\left(\dfrac{75° + 15°}{2}\right) \cos!\left(\dfrac{75° - 15°}{2}\right) = 2 \sin 45° \cos 30°.$$

Substitute special-angle values: $\sin 45° = \sqrt{2}/2$, $\cos 30° = \sqrt{3}/2$.

$$\sin 75° + \sin 15° = 2 \cdot \dfrac{\sqrt{2}}{2} \cdot \dfrac{\sqrt{3}}{2} = \dfrac{\sqrt{6}}{2}.$$

In radians, $75° + 15° = 5\pi/12 + \pi/12 = \pi/2$, so the half-sum is $\pi/4$, and the half-difference is $\pi/6$.

Final answer: $\sin 75° + \sin 15° = \dfrac{\sqrt{6}}{2}$.

Standard (Wrong Path First — Quick — Standard — Stretch). Evaluate $\cos 75° - \cos 15°$ exactly.

The wrong path. A student remembers "$\cos$ minus $\cos$ gives sin times sin" and writes:

$$\cos 75° - \cos 15° = 2 \sin!\left(\dfrac{75° + 15°}{2}\right) \sin!\left(\dfrac{75° - 15°}{2}\right).$$

That has the right functions (sin and sin) but is missing the leading negative sign. The cos − cos identity carries a $-2$ in front, not a $+2$. The student would compute $+\sqrt{6}/2$ instead of the correct $-\sqrt{6}/2$.

The flaw: the $\cos A - \cos B$ identity has a negative coefficient. All four identities have a leading $2$ or $-2$ — the negative sign is unique to the cos-minus-cos case, and the most common slip is to drop it because the other three identities are all positive-leading.

The rescue. Apply the correct identity:

$$\cos A - \cos B = -2 \sin!\left(\dfrac{A+B}{2}\right) \sin!\left(\dfrac{A-B}{2}\right).$$

$$\cos 75° - \cos 15° = -2 \sin 45° \sin 30° = -2 \cdot \dfrac{\sqrt{2}}{2} \cdot \dfrac{1}{2} = -\dfrac{\sqrt{2}}{2}.$$

In radians, $\sin(\pi/4) \cdot \sin(\pi/6) = (\sqrt{2}/2)(1/2)$ — same arithmetic.

Sanity check: $\cos 75° \approx 0.2588$, $\cos 15° \approx 0.9659$, so $\cos 75° - \cos 15° \approx -0.7071 \approx -\sqrt{2}/2$ ✓.

Final answer: $\cos 75° - \cos 15° = -\dfrac{\sqrt{2}}{2}$.

In the McKinney TX Grade 11 cohort, the missing leading negative on $\cos - \cos$ is the most consistent first-attempt slip — roughly five out of every ten students drop the sign on the topic-introduction worksheet, even after seeing all four identities side by side.

Stretch. Express $\sin x + \sin 3x + \sin 5x + \sin 7x$ as a product.

Pair the outer terms and the inner terms:

$$(\sin x + \sin 7x) + (\sin 3x + \sin 5x).$$

Apply sum-to-product to each pair:

• $\sin x + \sin 7x = 2 \sin!\left(\dfrac{x + 7x}{2}\right) \cos!\left(\dfrac{x - 7x}{2}\right) = 2 \sin 4x \cos(-3x) = 2 \sin 4x \cos 3x$ (cosine is even).

• $\sin 3x + \sin 5x = 2 \sin 4x \cos(-x) = 2 \sin 4x \cos x$.

Sum the two pieces:

$$2 \sin 4x \cos 3x + 2 \sin 4x \cos x = 2 \sin 4x (\cos 3x + \cos x).$$

Now apply sum-to-product to $\cos 3x + \cos x$:

$$\cos 3x + \cos x = 2 \cos!\left(\dfrac{3x + x}{2}\right) \cos!\left(\dfrac{3x - x}{2}\right) = 2 \cos 2x \cos x.$$

So the full sum equals:

$$2 \sin 4x \cdot 2 \cos 2x \cos x = 4 \sin 4x \cos 2x \cos x.$$

In radians, the variable is $x$ throughout; in degrees, the variable could equally be written as $x°$.

Final answer: $\sin x + \sin 3x + \sin 5x + \sin 7x = 4 \sin 4x \cos 2x \cos x$.

A four-term sum collapsed to a four-factor product — exactly the kind of simplification that makes JEE / boards problems tractable.

Why the Sum to Product Identities Matter Outside the Classroom

The sum-to-product identities are the algebraic engine behind almost every wave-interference calculation in science and engineering.

• Acoustic beats. When two pure tones at frequencies $f_1$ and $f_2$ play together, the resulting waveform is $\sin(2\pi f_1 t) + \sin(2\pi f_2 t) = 2 \sin(2\pi \overline{f} t) \cos(2\pi \Delta f t)$ where $\overline{f}$ is the average frequency and $\Delta f$ is half the difference. The slow $\cos(2\pi \Delta f t)$ factor is what musicians hear as the "wow-wow-wow" beat envelope when tuning instruments.

• Radar and sonar — Doppler shift detection. A returned radar pulse beats against the transmitted pulse; the beat frequency, isolated via the sum-to-product identity, is the Doppler shift that gives the target's radial velocity.

• Quantum mechanics — wave packet construction. A Gaussian wave packet is built by summing many sinusoids; the sum-to-product identities are the bookkeeping for the constructive and destructive interference that produces the packet's shape.

• Audio synthesis — ring modulation. A ring modulator computes $\sin A \cdot \cos B$ — a product of trig functions — which by the product-to-sum direction equals $\tfrac{1}{2}[\sin(A+B) + \sin(A-B)]$, producing the characteristic metallic timbre used in Doctor Who's Dalek voices.

• Electrical engineering — power factor in AC circuits. Computing real power from voltage and current sinusoids involves $V(t) I(t)$ — a product — converted to sums via the product-to-sum identity to extract the DC (time-average) component.

The sum to product formula family lives at the boundary between wave addition and wave multiplication — exactly the calculation every interference effect requires.

A Brief History of Sum to Product

François Viète (1540–1603, France) was the first European mathematician to write down the modern sum-to-product identities in algebraic form, in his Canon Mathematicus (1579). The identities had been used implicitly by Hipparchus and Ptolemy for chord-table interpolation — but Viète gave them their first formal algebraic statement.

John Napier (1550–1617, Scotland) — the inventor of logarithms — used the sum-to-product identities as the core of his prosthaphaeresis technique: a way to multiply two numbers by first taking their cosines, applying the product-to-sum formula, and adding the results. The technique reduced 16th-century astronomical computation from days to hours, until Napier's own logarithm tables (1614) replaced it.

The story worth telling — Tycho Brahe (1546–1601, Denmark). Brahe ran the most accurate astronomical observatory in 16th-century Europe. To compute planetary positions from raw observations, he needed to multiply long decimal numbers — a punishing task with pen-and-ink. Brahe's solution was prosthaphaeresis — Greek for "addition and subtraction." To compute $0.71438 \times 0.27654$, Brahe's clerks looked up the angles whose cosines were those values, applied the sum-to-product formula $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$, added the right-hand-side cosines, and divided by 2.

The full multiplication became two table lookups, two additions, and one division — vastly cheaper than the direct multiplication. Tycho's observations of Mars were enabled by sum-to-product identities; those observations gave Johannes Kepler the data to derive his three laws of planetary motion. Without the sum-to-product formula, modern celestial mechanics would have begun a generation later.

Where Students Trip Up on Sum to Product

1. Forgetting the leading negative on $\cos A - \cos B$

Where it slips in: Three of the four identities have a leading $+2$; the fourth has $-2$. Students drop the negative.

Don't do this: Write $\cos A - \cos B = 2 \sin((A+B)/2) \sin((A-B)/2)$.

The correct way: $\cos A - \cos B = -2 \sin((A+B)/2) \sin((A-B)/2)$. The negative sign is the diagnostic.

2. Mixing up the half-sum and half-difference inside the functions

Where it slips in: A student writes $\sin A + \sin B = 2 \sin((A-B)/2) \cos((A+B)/2)$ — swapping the half-sum and half-difference.

Don't do this: Plug the angles in either order without checking.

The correct way: The half-sum $(A+B)/2$ goes inside the left function (the one that pairs with the parity of the original — sin for sin-sum, cos for cos-sum). The half-difference $(A-B)/2$ goes inside the right function. Verify by setting $A = B$: the formula should collapse to $2 \sin A$ when both are equal, which only works with the correct ordering.

3. Treating the formulas like $\sin(A+B) = \sin A + \sin B$

Where it slips in: A student writes $\sin A + \sin B = \sin(A + B)$ — the most fundamental trig confusion.

Don't do this: Distribute the function over the sum.

The correct way: $\sin$ does NOT distribute over addition. $\sin A + \sin B \ne \sin(A + B)$ in general. The sum-to-product formula is what correctly expresses $\sin A + \sin B$ — as a product, not a single trig function of the sum.

4. Applying sum-to-product to a sin + cos sum

Where it slips in: A student tries to use sum-to-product on $\sin x + \cos x$.

Don't do this: Apply the sin-plus-sin formula to a sin-plus-cos sum.

The correct way: The sum-to-product identities work for sin + sin or cos + cos (or differences). For $\sin x + \cos x$, rewrite as $\sqrt{2} \sin(x + \pi/4)$ via auxiliary-angle method — that's a different identity family.

Conclusion

• The sum to product formula family is four identities that rewrite $\sin A \pm \sin B$ and $\cos A \pm \cos B$ as products of trig functions of half-sum and half-difference angles.

• The half-sum $(A+B)/2$ goes inside the left function; the half-difference $(A-B)/2$ goes inside the right function.

• The single sign anomaly is $\cos A - \cos B$, which has a leading $-2$ — the others all have $+2$.

• They are proved by applying the sum-and-difference identities with the substitution $X = (A+B)/2$, $Y = (A-B)/2$.

• These identities power acoustic beat detection, radar Doppler analysis, FM synthesis, AC power factor calculations, and the matched-filter computation that detected gravitational waves at LIGO.

Try It Yourself — Three Sum to Product Problems

1. Express $\cos 50° + \cos 70°$ as a product, then evaluate.

2. Show that $\dfrac{\sin 5x - \sin x}{\cos 5x + \cos x} = \tan 2x$.

3. Solve for $x$ in $[0, 2\pi)$: $\sin 3x + \sin x = 0$.

If Problem 3 yields $x = 0, \pi/4, \pi/2, 3\pi/4, \pi, \dots$ — that's the answer set when $\sin 2x \cos x = 0$, factored via sum-to-product.

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