Sum-to-Product vs Product-to-Sum: Choose the Right Identity

You see sin 5x + sin 3x and remember that there is a formula somewhere. But should you use sum-to-product, an angle-addition formula, or something else entirely?

The quickest way to decide is to look at the shape of the expression and what you want to do with it. Two trigonometric terms being added or subtracted can often become one product. Two trigonometric factors being multiplied can often become a sum or difference. Neither direction is automatically better; the useful direction is the one that reveals the next step.

For example, sin 5x + sin 3x becomes 2 sin 4x cos x. That is helpful if you need to solve an equation or simplify a matching fraction. Meanwhile, sin 3x cos x becomes (sin 4x + sin 2x)/2, which is helpful when the question asks for an expression in separate angles.

Let’s build a reliable way to choose and check each formula.

A One-Minute Decision Map

What you see	First direction to try	Why it may help
sin A + sin B or sin A - sin B	Sum-to-product	Exposes a sine factor and a cosine factor
cos A + cos B or cos A - cos B	Sum-to-product	Exposes two factors, often useful for equations
sin A cos B or cos A sin B	Product-to-sum	Separates the angles into two sine terms
sin A sin B or cos A cos B	Product-to-sum	Separates the angles into cosine terms
A fraction with similar two-term sums above and below	Usually sum-to-product in both	May expose a common factor
A trigonometric equation equal to zero	Usually sum-to-product	A product equals zero when a factor equals zero

This is a starting point, not a rule that every question must obey. Sometimes an angle-addition or double-angle identity is shorter. The best first move depends on the destination.

The Two Angles Hidden Inside Every Sum

In the sum-to-product identities, the same two angles appear again and again:

M = (A + B)/2      the average angle
D = (A - B)/2      half the difference

They fit together neatly:

M + D = A
M - D = B

Think of M as the midpoint of the two angles and D as the step from that midpoint to either angle. If A = 70° and B = 30°, then M = 50° and D = 20°. The original angles are 50° + 20° and 50° - 20°.

That average-and-difference pattern explains why halving is essential. Using A + B and A - B without dividing by 2 is a common mistake.

The Four Sum-to-Product Identities

Keep M = (A + B)/2 and D = (A - B)/2 in view:

sin A + sin B =  2 sin M cos D
sin A - sin B =  2 cos M sin D

cos A + cos B =  2 cos M cos D
cos A - cos B = -2 sin M sin D

The order of the factors does not matter, so 2 cos M sin D is the same as 2 sin D cos M. The order of A and B, however, matters in a difference because D = (A - B)/2 changes sign when you swap them.

Here is a memory aid that is safer than learning four unrelated strings:

• Adding sines gives sin M cos D; subtracting sines gives cos M sin D.
• Adding cosines gives cos M cos D.
• Subtracting cosines gives negative sin M sin D.
• Every formula starts with a factor of 2.

You can also verify the signs by trying A = B. Both differences must become zero because D = 0, and sin D = 0. The cosine-sum identity becomes 2 cos A, because M = A and cos D = 1.

Example 1: Turn a sine sum into factors

Convert sin 5x + sin 3x.

A = 5x, B = 3x
M = (5x + 3x)/2 = 4x
D = (5x - 3x)/2 = x

sin 5x + sin 3x = 2 sin 4x cos x

Notice that 4x sits midway between 5x and 3x.

Example 2: The cosine-difference sign

Convert cos 5x - cos 3x.

M = (5x + 3x)/2 = 4x
D = (5x - 3x)/2 = x

cos 5x - cos 3x = -2 sin 4x sin x

The minus sign is part of the identity. A quick check at a small positive value of x can catch an incorrect plus sign: cosine decreases near zero, so cos 5x - cos 3x is negative there, just like -2 sin 4x sin x.

Example 3: Exact value without finding each term

Find sin 75° + sin 15°.

M = (75° + 15°)/2 = 45°
D = (75° - 15°)/2 = 30°

sin 75° + sin 15°
= 2 sin 45° cos 30°
= 2(√2/2)(√3/2)
= √6/2

You did not need to calculate sin 75° and sin 15° separately. The average and half-difference landed on familiar angles.

Example 4: A difference with the smaller angle first

Find cos 20° - cos 80°.

M = 50°
D = (20° - 80°)/2 = -30°

cos 20° - cos 80°
= -2 sin 50° sin(-30°)
= sin 50°

The two negative signs cancel. Do not silently replace D = -30° with 30°; make the oddness of sine visible in the working.

The Product-to-Sum Identities

When you start with multiplication, the new angles are A + B and A - B. There is a factor of 1/2 in every formula:

sin A cos B = [sin(A + B) + sin(A - B)]/2
cos A sin B = [sin(A + B) - sin(A - B)]/2

cos A cos B = [cos(A + B) + cos(A - B)]/2
sin A sin B = [cos(A - B) - cos(A + B)]/2

The mixed products look almost identical, but swapping sine and cosine changes the sign before sin(A - B). That is because sin(B - A) = -sin(A - B).

The same-function products are easier to distinguish:

• Cosine times cosine gives a sum of two cosines.
• Sine times sine gives the difference of two cosines, with the difference angle first.

Example 5: The swapped mixed product

Convert cos 5x sin 2x.

cos 5x sin 2x
= [sin(5x + 2x) - sin(5x - 2x)]/2
= (sin 7x - sin 3x)/2

You could write the sine factor first instead: sin 2x cos 5x. Then the formula produces sin(2x + 5x) + sin(2x - 5x), divided by 2. Since sin(-3x) = -sin 3x, the answer is the same. This is a useful self-check.

Example 6: Sine times sine

Convert sin 5x sin 2x.

sin 5x sin 2x
= [cos(5x - 2x) - cos(5x + 2x)]/2
= (cos 3x - cos 7x)/2

Do not reverse the final subtraction. For a quick check, let x = 0. Both sides are zero. To test the sign, choose a nonzero angle, or derive the formula from the cosine addition identities below.

Example 7: An exact product

Find sin 75° sin 15°.

sin 75° sin 15°
= [cos(75° - 15°) - cos(75° + 15°)]/2
= [cos 60° - cos 90°]/2
= (1/2 - 0)/2
= 1/4

The same pair of angles as Example 3 gives a different task and a different route. The operation between the terms tells you which direction to take.

Why These Formulas Work

You do not have to treat the eight identities as eight unrelated facts. They grow from the angle-addition identities:

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

Add these two lines. The cos A sin B terms cancel:

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

Divide by 2 and you have the product-to-sum formula for sin A cos B. Subtract the second line from the first instead, and you get the formula for cos A sin B.

Likewise:

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

Adding gives 2 cos A cos B; subtracting the first line from the second gives 2 sin A sin B. This also confirms the order cos(A - B) - cos(A + B).

To travel in the reverse direction, call the original angles u and v, and set:

A = (u + v)/2
B = (u - v)/2

Then A + B = u and A - B = v. Substituting these into the product-to-sum formulas gives the sum-to-product formulas. The two families are genuinely two directions through the same doorway.

Choosing the Direction in a Longer Question

When solving an equation, expose factors

Solve sin 5x - sin x = 0, with x measured in radians.

sin 5x - sin x = 2 cos 3x sin 2x

So 2 cos 3x sin 2x = 0.

At least one factor must be zero:

sin 2x = 0  gives  x = nπ/2

cos 3x = 0  gives  x = π/6 + nπ/3

Here n is any integer. Both families matter; stopping after the first one would miss solutions. In an interval question, keep only values in the stated interval and remove repeats.

When simplifying a fraction, factor both parts

Consider:

(sin 5x + sin 3x)/(cos 5x + cos 3x)

Using sum-to-product in the numerator and denominator:

sin 5x + sin 3x = 2 sin 4x cos x
cos 5x + cos 3x = 2 cos 4x cos x

Therefore the fraction equals tan 4x wherever the original denominator is nonzero. More explicitly, the cancellation requires cos x != 0, and tan 4x requires cos 4x != 0. If cos x = 0, the original expression is 0/0 and is undefined, even though tan 4x might have a value there. A simplification must not quietly add new allowed values.

When angles are multiplied, separate them

Suppose you need to rewrite cos 5x cos 2x as a sum:

cos 5x cos 2x
= [cos(5x - 2x) + cos(5x + 2x)]/2
= (cos 3x + cos 7x)/2

This direction is also useful when matching the shape of the other side of an identity. If you are proving one, work from the more complicated side and choose the conversion that moves it closer to the target. For a broader proof strategy, see how to prove trigonometric identities without expanding the wrong side.

Five Checks That Catch Most Mistakes

1. Match the form first. sin(A + B) is one sine of a combined angle, not sin A + sin B. Use an angle-addition identity for the former and sum-to-product for the latter.
2. Keep units consistent. You can work in degrees or radians, but do not mix 30° with π/6 without converting.
3. Halve both combinations in sum-to-product. The angles are (A + B)/2 and (A - B)/2, not merely A + B and A - B.
4. Watch odd and even functions. sin(-t) = -sin t, while cos(-t) = cos t. A negative difference angle may change the sign of a sine term but not a cosine term.
5. Check a simple input. At A = B, both differences become zero. At B = 0, sin A cos 0 = sin A. Such checks can expose a missing 2, 1/2, or minus sign.

Practice: Choose Before You Calculate

Try each question before looking at the answers. Name the direction you are using.

1. Write sin 7x + sin 3x as a product.
2. Write sin 7x - sin 3x as a product.
3. Write cos 7x - cos 3x as a product.
4. Write cos 6x + cos 2x as a product.
5. Write sin 4x cos x as a sum.
6. Write cos 4x sin x as a sum.
7. Write sin 4x sin x as a difference.
8. Evaluate cos 75° cos 15° exactly.
9. Solve cos 3x + cos x = 0 for 0 <= x < 2π.
10. Simplify (sin 7x - sin 3x)/(cos 7x + cos 3x), stating where the original fraction is defined.

Answers and quick checks

1. 2 sin 5x cos 2x. The average is 5x and half-difference is 2x.
2. 2 cos 5x sin 2x.
3. -2 sin 5x sin 2x. Do not lose the leading minus.
4. 2 cos 4x cos 2x.
5. (sin 5x + sin 3x)/2.
6. (sin 5x - sin 3x)/2.
7. (cos 3x - cos 5x)/2.
8. [cos 60° + cos 90°]/2 = 1/4.
9. 2 cos 2x cos x = 0, so x = π/4, 3π/4, 5π/4, 7π/4 from cos 2x = 0, and x = π/2, 3π/2 from cos x = 0.
10. The fraction becomes tan 2x, provided cos 5x != 0 and cos 2x != 0. The original denominator is 2 cos 5x cos 2x; exclude all its zeros even if tan 2x would be defined there.

If the answers to 1 and 5 feel like mirror images, that is the point. Learning to travel both ways is more useful than memorising a list without recognising when to use it.

For additional worked examples and the underlying angle-addition identities, see OpenStax’s lesson on the two formula families and MIT’s short trigonometric-identity notes.

Frequently Asked Questions

What is the main difference between sum-to-product and product-to-sum?

Sum-to-product changes two sine or cosine terms joined by + or - into a multiplication of factors. Product-to-sum changes two multiplied trigonometric factors into two separate terms. They reverse the same angle patterns.

How do I decide which identity to use first?

Look at the present form and the goal. A sum or difference often calls for sum-to-product when you want factors, especially in equations and fractions. A product often calls for product-to-sum when you need separate angle terms. If the target suggests an angle-addition identity instead, use that.

Do I always divide the angles by 2?

Only in sum-to-product. Starting with sin A + sin B, for example, you use (A + B)/2 and (A - B)/2. Starting with sin A cos B, product-to-sum uses the full angles A + B and A - B, with 1/2 multiplying the resulting expression.

Why does cos A - cos B have a minus sign?

Because subtracting the cosine angle-addition equations gives cos(A + B) - cos(A - B) = -2 sin A sin B. Replacing those two combined angles by the original pair produces the negative sign in the cosine-difference formula.

What if the smaller angle is written first in a difference?

Keep the given order in (A - B)/2. A negative half-difference is fine. Then use sin(-t) = -sin t or cos(-t) = cos t if you want a positive angle in the final answer.

Are sin A cos B and cos A sin B the same product?

Not generally. Multiplication lets you reorder the same two factors, such as sin A cos B = cos B sin A. But cos A sin B uses different functions on the angles. Its product-to-sum formula has the opposite sign before sin(A - B).

Can I use these identities with radians?

Yes. The identities work in degrees and radians. Keep all angles in one unit while adding, subtracting, and halving them.

Can a sum-to-product answer look different from the one in a book?

Yes. Factor order can change, and a negative sign may move inside a sine. For instance, -2 sin M sin((A - B)/2) equals 2 sin M sin((B - A)/2). Check whether the expressions are equivalent before treating one as wrong.

Does factoring always solve a trigonometric equation completely?

Factoring reveals the separate zero-factor cases, but you still need to solve every case, apply the stated interval, and remove repeated solutions. In a fraction, you must also retain the original restrictions on the denominator.

Is a quick substitution enough to prove an identity?

No. Trying a convenient angle can expose a mistake, but agreement at one or several values is not proof. Derive the identity from the angle-addition formulas, or transform one side into the other for all allowed angles.

What is a simple way to remember the product-to-sum signs?

Cosine times cosine gives a sum of cosines. Sine times sine gives cos(A - B) - cos(A + B). For mixed products, write the sine factor’s angle first and use sin A cos B = [sin(A + B) + sin(A - B)]/2; any negative angle can then be simplified carefully.

Which formulas should I learn first if I am short on time?

Start with the sine and cosine angle-addition identities and understand how adding or subtracting them produces products. Then learn the average-angle and half-difference pattern for the reverse direction. This gives you a way to rebuild a forgotten sign instead of guessing it.