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Precalculus · F.TF.C.9

Product to sum and sum to product formulas

Neither 52.5^\circ nor 7.5^\circ is a special angle. Can \sin 52.5^\circ\cos 7.5^\circ still be found exactly from special-angle values, and can a sum of two waves be read as one wave times another?

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Cosine of a sum or differencePrecalculus
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Product to sum and sum to product formulas
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Why this exists

A product of two trig functions is awkward. To find \sin 52.5^\circ\cos 7.5^\circ exactly you would need each factor, and neither angle is on the unit circle’s list of special angles. To graph \cos 5x\cos 2x you cannot simply add two familiar waves. A sum has the opposite problem: the graph of \cos 9t + \cos 7t is hard to picture, and an equation such as \sin 5x + \sin x = 0 cannot be split into simpler equations until the left side is a product.

The cosine sum and difference formulas and the sine versions already contain the fix. Each pair has the same two products on the right with opposite signs, so adding or subtracting a pair isolates one product. That gives formulas that turn a product into a sum, and read backward, formulas that turn a sum into a product.

Definition

For all real numbers a and b (or angles in degrees), the product to sum formulas are

\begin{aligned} \cos a\cos b &= \tfrac{1}{2}\left[\cos(a - b) + \cos(a + b)\right] \\ \sin a\sin b &= \tfrac{1}{2}\left[\cos(a - b) - \cos(a + b)\right] \\ \sin a\cos b &= \tfrac{1}{2}\left[\sin(a + b) + \sin(a - b)\right] \end{aligned}

A product of two cosines, of two sines, or of a sine and a cosine becomes half a sum or difference of single functions, with no products left.

For all real numbers u and v, the sum to product formulas are

\begin{aligned} \sin u + \sin v &= 2\sin\tfrac{u + v}{2}\cos\tfrac{u - v}{2} \\ \sin u - \sin v &= 2\cos\tfrac{u + v}{2}\sin\tfrac{u - v}{2} \\ \cos u + \cos v &= 2\cos\tfrac{u + v}{2}\cos\tfrac{u - v}{2} \\ \cos u - \cos v &= -2\sin\tfrac{u + v}{2}\sin\tfrac{u - v}{2} \end{aligned}

Both lists are identities in the sense of What an identity is: true for every value, with nothing excluded. The angles in a sum to product formula are the half sum \displaystyle \frac{u + v}{2} and the half difference \displaystyle \frac{u - v}{2}.

Intuition

t y 2 −2
\displaystyle \frac{\pi}{2}
\pi
\displaystyle \frac{3\pi}{2}
2\pi
Figure 1The solid curve is f(t) = \cos 9t + \cos 7t, which equals 2\cos 8t\cos t. The dashed curves are y = 2\cos t and y = -2\cos t; the fast wave stays between them and dies out where \cos t = 0.

Read the cosine sum formula backward: \cos u + \cos v is a product of the cosine of the average angle and the cosine of half the gap. For two waves whose angles grow at nearly the same rate, such as \cos 9t and \cos 7t, the average 8t grows fast and half the gap, t, grows slowly. So the sum is a fast wave, \cos 8t, whose height is set by a slow wave, 2\cos t.

The picture agrees with what the two waves do. At t = 0 both equal 1 and the sum is 2. As t grows, 9t pulls ahead of 7t. At \displaystyle t = \frac{\pi}{2} it is ahead by exactly \pi, half a turn, and \cos(\theta + \pi) = -\cos\theta, so the two waves cancel. At t = \pi it is ahead by a full turn, the waves are back in step, and the sum again reaches size 2. The dashed curves y = \pm 2\cos t trace that slow swell.

Results

Product to sum. For all real a and b,

\cos a\cos b = \tfrac{1}{2}\left[\cos(a - b) + \cos(a + b)\right]
\sin a\sin b = \tfrac{1}{2}\left[\cos(a - b) - \cos(a + b)\right]
\sin a\cos b = \tfrac{1}{2}\left[\sin(a + b) + \sin(a - b)\right]

A product \cos a\sin b needs no fourth formula: it is \sin b\cos a, the third formula with the letters traded (Exercise 20).

Sum to product. For all real u and v,

\sin u + \sin v = 2\sin\tfrac{u + v}{2}\cos\tfrac{u - v}{2}
\sin u - \sin v = 2\cos\tfrac{u + v}{2}\sin\tfrac{u - v}{2}
\cos u + \cos v = 2\cos\tfrac{u + v}{2}\cos\tfrac{u - v}{2}
\cos u - \cos v = -2\sin\tfrac{u + v}{2}\sin\tfrac{u - v}{2}

Equal angles. With b = a the third product formula gives \displaystyle \sin a\cos a = \frac{1}{2}\sin 2a, the double angle formula, and the first two give the Power reducing and half angle formulas (Exercise 23).

Why it is true

Product to sum. Write the two cosine formulas one above the other:

\begin{aligned} \cos(a - b) &= \cos a\cos b + \sin a\sin b \\ \cos(a + b) &= \cos a\cos b - \sin a\sin b \end{aligned}

Adding them cancels the \sin a\sin b terms: \cos(a - b) + \cos(a + b) = 2\cos a\cos b. Subtracting the second from the first cancels the \cos a\cos b terms: \cos(a - b) - \cos(a + b) = 2\sin a\sin b. Dividing each by 2 gives the first two formulas. The sine formulas

\begin{aligned} \sin(a + b) &= \sin a\cos b + \cos a\sin b \\ \sin(a - b) &= \sin a\cos b - \cos a\sin b \end{aligned}

add to \sin(a + b) + \sin(a - b) = 2\sin a\cos b, and dividing by 2 gives the third.

Sum to product. Each line just proved has the shape (sum of two functions) = 2 \times (a product). Rename the angles: let u = a + b and v = a - b. Adding and subtracting these gives

a = \frac{u + v}{2}, \qquad b = \frac{u - v}{2}

so every pair u, v comes from exactly one pair a, b. Substituting into \sin(a + b) + \sin(a - b) = 2\sin a\cos b gives the first sum to product formula. Substituting into \cos(a - b) + \cos(a + b) = 2\cos a\cos b gives \cos v + \cos u on the left, which is the third. Substituting into \cos(a - b) - \cos(a + b) = 2\sin a\sin b gives \displaystyle \cos v - \cos u = 2\sin\frac{u + v}{2}\sin\frac{u - v}{2}; multiplying by -1 gives the fourth. For the second, replace v by -v in the first and use that sine is odd and cosine is even (Unit circle symmetry and even-odd identities):

\begin{aligned} \sin u - \sin v &= \sin u + \sin(-v) \\ &= 2\sin\tfrac{u - v}{2}\cos\tfrac{u + v}{2} \end{aligned}

which is the second formula with its factors reordered (multiplication is commutative).

Worked examples

Example 1

Products of two cosines and of two sines

Write each as a sum or difference with no products: (a) 4\cos 5x\cos 2x; (b) 6\sin 3x\sin 7x.

Solution

(a) The first product formula with a = 5x and b = 2x gives a - b = 3x and a + b = 7x:

\begin{aligned} 4\cos 5x\cos 2x &= 4\cdot\tfrac{1}{2}\left[\cos 3x + \cos 7x\right] \\ &= 2\cos 3x + 2\cos 7x \end{aligned}

(b) The second formula with a = 3x and b = 7x gives a - b = -4x, and cosine is even, so \cos(-4x) = \cos 4x:

\begin{aligned} 6\sin 3x\sin 7x &= 6\cdot\tfrac{1}{2}\left[\cos(-4x) - \cos 10x\right] \\ &= 3\cos 4x - 3\cos 10x \end{aligned}

Check (b) at \displaystyle x = \frac{\pi}{2}: the product is \displaystyle 6\sin\frac{3\pi}{2}\sin\frac{7\pi}{2} = 6(-1)(-1) = 6, and 3\cos 2\pi - 3\cos 5\pi = 3 + 3 = 6.

Example 2

Laura’s exact product

Laura wants the exact value of \sin 52.5^\circ\cos 7.5^\circ.

Solution

Neither angle is a special angle, but the sum and difference of the angles are: 52.5^\circ + 7.5^\circ = 60^\circ and 52.5^\circ - 7.5^\circ = 45^\circ. The third product formula gives

\begin{aligned} \sin 52.5^\circ\cos 7.5^\circ &= \tfrac{1}{2}\left[\sin 60^\circ + \sin 45^\circ\right] \\ &= \tfrac{1}{2}\left[\tfrac{\sqrt{3}}{2} + \tfrac{\sqrt{2}}{2}\right] \\ &= \frac{\sqrt{3} + \sqrt{2}}{4} \end{aligned}

Check. A calculator gives \sin 52.5^\circ \approx 0.7934 and \cos 7.5^\circ \approx 0.9914, whose product is about 0.7866, and \displaystyle \frac{\sqrt{3} + \sqrt{2}}{4} \approx 0.7866.

Example 3

Sums written as products

Write each as a product: (a) \sin 7x + \sin 3x; (b) \cos x - \cos 5x.

Solution

(a) The half sum is \displaystyle \frac{7x + 3x}{2} = 5x and the half difference is \displaystyle \frac{7x - 3x}{2} = 2x, so

\sin 7x + \sin 3x = 2\sin 5x\cos 2x

(b) With u = x and v = 5x the half sum is 3x and the half difference is -2x. The fourth formula and the oddness of sine, \sin(-2x) = -\sin 2x, give

\begin{aligned} \cos x - \cos 5x &= -2\sin 3x\sin(-2x) \\ &= 2\sin 3x\sin 2x \end{aligned}

Check (b) at \displaystyle x = \frac{\pi}{4}: \displaystyle \cos\frac{\pi}{4} - \cos\frac{5\pi}{4} = \sqrt{2}, and \displaystyle 2\sin\frac{3\pi}{4}\sin\frac{\pi}{2} = 2\cdot\frac{\sqrt{2}}{2}\cdot 1 = \sqrt{2}.

Example 4

A sum of three cosines that is zero

Show that \cos 40^\circ + \cos 80^\circ + \cos 160^\circ = 0.

Solution

None of the three angles is a special angle, so pair two of the terms. The first two have half sum 60^\circ and half difference -20^\circ:

\begin{aligned} \cos 40^\circ + \cos 80^\circ &= 2\cos 60^\circ\cos(-20^\circ) \\ &= 2\cdot\tfrac{1}{2}\cos 20^\circ = \cos 20^\circ \end{aligned}

The third is \cos 160^\circ = \cos(180^\circ - 20^\circ) = -\cos 20^\circ, by the reflection across the y-axis in Unit circle symmetry and even-odd identities. So the whole sum is \cos 20^\circ - \cos 20^\circ = 0. A calculator agrees: 0.76604 + 0.17365 - 0.93969 = 0.00000.

Example 5

Ulzhan’s identity

Ulzhan verifies that \dfrac{\sin 5t + \sin 3t}{\cos 5t + \cos 3t} = \tan 4t wherever the left side is defined.

Solution

Work on the left side only, as in Verifying identities. Both sums have half sum 4t and half difference t:

\begin{aligned} \frac{\sin 5t + \sin 3t}{\cos 5t + \cos 3t} &= \frac{2\sin 4t\cos t}{2\cos 4t\cos t} \\ &= \frac{\sin 4t}{\cos 4t} = \tan 4t \end{aligned}

Cancelling 2\cos t is allowed because the left side is defined only where its denominator 2\cos 4t\cos t is not 0, so \cos t \neq 0 there. The identity holds on that domain and no further: at \displaystyle t = \frac{\pi}{2} the right side is \tan 2\pi = 0, but the left side is \displaystyle \frac{0}{0}, undefined.

Example 6

Elan’s slow swell

Elan graphs f(t) = \cos 9t + \cos 7t for 0 \le t \le 2\pi (the figure above). Write f(t) as a product, explain why the graph stays between y = 2\cos t and y = -2\cos t, and find where the swell dies out and the value f(\pi).

Solution

The half sum is 8t and the half difference is t, so

f(t) = 2\cos 8t\cos t

Since |\cos 8t| \le 1, the size of f(t) is at most |2\cos t|, so the graph lies between the two dashed curves. It touches one of them exactly where \cos 8t = \pm 1, at \displaystyle t = \frac{k\pi}{8} for integers k. The factor \cos 8t has period \displaystyle \frac{2\pi}{8} = \frac{\pi}{4} and makes the fast wiggles; the factor 2\cos t has period 2\pi and sets their height.

The swell dies out where the slow factor is 0: \cos t = 0 at \displaystyle t = \frac{\pi}{2} and \displaystyle t = \frac{3\pi}{2}. There f = 0 whatever \cos 8t is. At t = \pi, f(\pi) = 2\cos 8\pi\cos\pi = 2(1)(-1) = -2, the deepest the graph goes. Directly, \cos 9\pi + \cos 7\pi = -1 - 1 = -2.

Boundaries

  • Every formula holds for all real a, b, u, v, in radians or in degrees, as long as all the angles in one formula use the same unit.
  • An exact value needs the new angles to be special. \sin 52.5^\circ\cos 7.5^\circ is exact because 60^\circ and 45^\circ are; \sin 50^\circ\cos 7^\circ obeys the same formula but gives nothing exact.
  • The sum to product formulas join two sines or two cosines with coefficient 1 each. A mixed sum such as \sin u + \cos v must first become two sines, for example through \displaystyle \cos v = \sin\left(\frac{\pi}{2} - v\right), and a sum such as 3\sin u + \sin v does not factor this way. A sine plus a cosine of the same angle is a different problem, Combining a sine and a cosine into one wave.
  • An identity reached by cancelling a factor holds only where that factor is not 0, as in Ulzhan’s identity at \displaystyle t = \frac{\pi}{2}.

Where this goes next

Turning a sum into a product is the first move in Solving trig equations by factoring, quadratics and identities: once \sin 5x + \sin x is 2\sin 3x\cos 2x, the equation \sin 5x + \sin x = 0 splits into two simple ones (Exercise 18), each a basic trig equation. Turning a product into a sum is what later lets a product of waves be handled one term at a time. The next lesson, Combining a sine and a cosine into one wave, writes a\sin t + b\cos t as a single wave, the case these formulas leave out. The whole family belongs to Trigonometric identities.

Exercises

Products to sums

  1. Write 2\cos 6x\cos x as a sum.
  2. Write 10\sin 4x\sin 9x as a difference.
  3. Write \sin 8t\cos 3t as a sum.
  4. Write \cos 2x\sin 6x as a sum.
  5. Write 4\sin(x + y)\sin(x - y) as a difference of cosines.

Exact values of products

  1. Find the exact value of \cos 37.5^\circ\cos 7.5^\circ.
  2. Find the exact value of \sin 67.5^\circ\sin 22.5^\circ.
  3. Find the exact value of \sin 142.5^\circ\cos 97.5^\circ.
  4. Find the exact value of \displaystyle \cos\frac{7\pi}{24}\cos\frac{\pi}{24}.

Sums to products

  1. Write \sin 9x + \sin 5x as a product.
  2. Write \cos 4t + \cos 10t as a product.
  3. Write \sin 2x - \sin 8x as a product.
  4. Write \cos 3x - \cos 11x as a product.
  5. Write \cos 85^\circ + \cos 35^\circ as a single cosine.
  6. Write \sin 100^\circ - \sin 20^\circ as a single sine.

Identities and equations

  1. Verify that \dfrac{\cos 3x - \cos 7x}{\sin 7x + \sin 3x} = \tan 2x wherever the left side is defined.
  2. Show that \cos x\cos 2x\cos 3x equals \displaystyle \frac{1}{4}\left(1 + \cos 2x + \cos 4x + \cos 6x\right).
  3. Solve \sin 5x + \sin x = 0 for x in [0, \pi).
  4. Laura graphs g(t) = \sin 11t - \sin 9t. Write g(t) as a product, and find every t in [0, \pi] where g(t) = 0.

Reasoning

  1. Starting from \displaystyle \sin a\cos b = \frac{1}{2}\left[\sin(a + b) + \sin(a - b)\right], derive a formula for \cos a\sin b, and explain why it needs no separate proof.
  2. Elan remembers the second formula as \displaystyle \sin a\sin b = \frac{1}{2}\left[\cos(a + b) - \cos(a - b)\right]. Test it at \displaystyle a = b = \frac{\pi}{2}, and explain which way round the cosines must go and why.
  3. The sum to product formulas came from the product to sum formulas by setting u = a + b and v = a - b. Explain why that substitution proves them for every pair u, v, not only for some.
  4. Put b = a in the first two product to sum formulas, and show that the results are \displaystyle \cos^2 a = \frac{1}{2}(1 + \cos 2a) and \displaystyle \sin^2 a = \frac{1}{2}(1 - \cos 2a).
  5. Ulzhan says that \cos u + \cos v = 0 whenever u - v is an odd multiple of \pi. Explain why, using a sum to product formula.
Answers
  1. \cos 5x + \cos 7x. The first formula with a = 6x, b = x; the 2 cancels the \displaystyle \frac{1}{2}.
  2. 5\cos 5x - 5\cos 13x. The second formula gives 5\left[\cos(-5x) - \cos 13x\right], and cosine is even.
  3. \displaystyle \frac{1}{2}\sin 11t + \frac{1}{2}\sin 5t. The third formula with a = 8t, b = 3t.
  4. \displaystyle \frac{1}{2}\sin 8x + \frac{1}{2}\sin 4x. Reorder to \sin 6x\cos 2x and use the third formula with a = 6x, b = 2x.
  5. 2\cos 2y - 2\cos 2x. The angles differ by (x + y) - (x - y) = 2y and add to 2x, so the product is \displaystyle 4\cdot\frac{1}{2}\left[\cos 2y - \cos 2x\right].
  6. \displaystyle \frac{\sqrt{3} + \sqrt{2}}{4}. The angles differ by 30^\circ and add to 45^\circ: \displaystyle \frac{1}{2}\left[\cos 30^\circ + \cos 45^\circ\right].
  7. \displaystyle \frac{\sqrt{2}}{4}. \displaystyle \frac{1}{2}\left[\cos 45^\circ - \cos 90^\circ\right] = \frac{1}{2}\cdot\frac{\sqrt{2}}{2}.
  8. \displaystyle \frac{\sqrt{2} - \sqrt{3}}{4}, about -0.0795. \displaystyle \frac{1}{2}\left[\sin 240^\circ + \sin 45^\circ\right] = \frac{1}{2}\left[-\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2}\right].
  9. \displaystyle \frac{\sqrt{2} + 1}{4}. The angles differ by \displaystyle \frac{6\pi}{24} = \frac{\pi}{4} and add to \displaystyle \frac{8\pi}{24} = \frac{\pi}{3}: \displaystyle \frac{1}{2}\left[\frac{\sqrt{2}}{2} + \frac{1}{2}\right].
  10. 2\sin 7x\cos 2x. Half sum 7x, half difference 2x.
  11. 2\cos 7t\cos 3t. Half sum 7t, half difference -3t, and \cos(-3t) = \cos 3t.
  12. -2\cos 5x\sin 3x. The second formula gives 2\cos 5x\sin(-3x), and sine is odd.
  13. 2\sin 7x\sin 4x. The fourth formula gives -2\sin 7x\sin(-4x), and sine is odd.
  14. \cos 25^\circ. \displaystyle 2\cos 60^\circ\cos 25^\circ = 2\cdot\frac{1}{2}\cos 25^\circ.
  15. \sin 40^\circ. \displaystyle 2\cos 60^\circ\sin 40^\circ = 2\cdot\frac{1}{2}\sin 40^\circ.
  16. The numerator is -2\sin 5x\sin(-2x) = 2\sin 5x\sin 2x and the denominator is 2\sin 5x\cos 2x. Where the left side is defined, 2\sin 5x\cos 2x \neq 0, so \sin 5x cancels, leaving \displaystyle \frac{\sin 2x}{\cos 2x} = \tan 2x.
  17. First \displaystyle \cos x\cos 3x = \frac{1}{2}(\cos 2x + \cos 4x). Multiplying by \cos 2x gives \displaystyle \frac{1}{2}\cos^2 2x + \frac{1}{2}\cos 4x\cos 2x. The first term is \displaystyle \frac{1}{4}(1 + \cos 4x) (the first formula with both angles 2x) and the second is \displaystyle \frac{1}{4}(\cos 2x + \cos 6x). Together they make the right side.
  18. \displaystyle x = 0, \frac{\pi}{4}, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{3\pi}{4}. The left side is 2\sin 3x\cos 2x. \sin 3x = 0 gives 3x = k\pi, so \displaystyle x = 0, \frac{\pi}{3}, \frac{2\pi}{3} in the interval; \cos 2x = 0 gives \displaystyle 2x = \frac{\pi}{2}, \frac{3\pi}{2}, so \displaystyle x = \frac{\pi}{4}, \frac{3\pi}{4}.
  19. g(t) = 2\cos 10t\sin t, and g(t) = 0 at t = 0, t = \pi and \displaystyle t = \frac{\pi}{20}, \frac{3\pi}{20}, \frac{5\pi}{20}, \ldots, \frac{19\pi}{20} (ten values). \sin t = 0 gives 0 and \pi; \cos 10t = 0 gives \displaystyle 10t = \frac{\pi}{2} + k\pi for integers k, so \displaystyle t = \frac{(2k + 1)\pi}{20} with k = 0, 1, \ldots, 9.
  20. \displaystyle \cos a\sin b = \frac{1}{2}\left[\sin(a + b) - \sin(a - b)\right]. Since \cos a\sin b = \sin b\cos a, the third formula with the letters traded gives \displaystyle \frac{1}{2}\left[\sin(b + a) + \sin(b - a)\right], and \sin(b - a) = -\sin(a - b) because sine is odd. The third formula holds for every pair of angles, so it holds for the pair b, a.
  21. At \displaystyle a = b = \frac{\pi}{2} the left side is 1 but Elan’s right side is \displaystyle \frac{1}{2}\left[\cos\pi - \cos 0\right] = -1. The correct form is \displaystyle \frac{1}{2}\left[\cos(a - b) - \cos(a + b)\right]: in \cos(a - b) the \sin a\sin b term carries a plus sign, so subtracting \cos(a + b) from \cos(a - b), not the other way, leaves +2\sin a\sin b.
  22. Given any real u and v, the numbers \displaystyle a = \frac{u + v}{2} and \displaystyle b = \frac{u - v}{2} are real and satisfy a + b = u and a - b = v. The product to sum formulas hold for that a and b, so the sum to product statement holds for that u and v. No pair is missed.
  23. With b = a, \cos(a - b) = \cos 0 = 1 and \cos(a + b) = \cos 2a. The first formula becomes \displaystyle \cos^2 a = \frac{1}{2}(1 + \cos 2a) and the second \displaystyle \sin^2 a = \frac{1}{2}(1 - \cos 2a), the Power reducing and half angle formulas.
  24. \displaystyle \cos u + \cos v = 2\cos\frac{u + v}{2}\cos\frac{u - v}{2}. If u - v = (2k + 1)\pi for an integer k, then \displaystyle \frac{u - v}{2} = \frac{\pi}{2} + k\pi, where cosine is 0, so the product is 0 whatever u + v is.
kg1128 · updated 2026-10-20 · CC-BY-SA-4.0