Mathwords: Product to Sum Identities

Key Formula

\sin A \cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)]

\cos A \sin B = \tfrac{1}{2}[\sin(A+B) - \sin(A-B)]

\cos A \cos B = \tfrac{1}{2}[\cos(A-B) + \cos(A+B)]

\sin A \sin B = \tfrac{1}{2}[\cos(A-B) - \cos(A+B)]

Where:

$A$ = The angle (or expression) in the first trigonometric factor
$B$ = The angle (or expression) in the second trigonometric factor

Worked Example

Problem: Rewrite sin(75°)cos(15°) as a sum of trigonometric functions.

Step 1: Identify which identity to use. We have sin times cos, so use the identity for sin A cos B.

\sin A \cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)]

Step 2: Assign the angles: A = 75° and B = 15°.

A = 75°, \quad B = 15°

Step 3: Compute A + B and A − B.

A + B = 90°, \quad A - B = 60°

Step 4: Substitute into the identity.

\sin 75° \cos 15° = \tfrac{1}{2}[\sin 90° + \sin 60°]

Step 5: Evaluate the known values: sin 90° = 1 and sin 60° = √3/2.

= \tfrac{1}{2}\!\left[1 + \frac{\sqrt{3}}{2}\right] = \tfrac{1}{2} + \frac{\sqrt{3}}{4} = \frac{2 + \sqrt{3}}{4}

Answer: sin 75° cos 15° = (2 + √3)/4

Another Example

This example uses variable expressions instead of specific degree values, and it applies the sin·sin identity (which produces cosines) rather than the sin·cos identity used in the first example.

Problem: Rewrite sin(5x)sin(3x) as a sum or difference of cosines.

Step 1: We have a product of two sines, so use the sin A sin B identity.

\sin A \sin B = \tfrac{1}{2}[\cos(A-B) - \cos(A+B)]

Step 2: Assign A = 5x and B = 3x.

A = 5x, \quad B = 3x

Step 3: Compute A − B and A + B.

A - B = 2x, \quad A + B = 8x

Step 4: Substitute into the identity.

\sin(5x)\sin(3x) = \tfrac{1}{2}[\cos(2x) - \cos(8x)]

Answer: sin(5x)sin(3x) = ½[cos(2x) − cos(8x)]

Frequently Asked Questions

How do you derive the product to sum identities?

They come directly from the angle addition and subtraction formulas. For example, if you write out sin(A+B) = sin A cos B + cos A sin B and sin(A−B) = sin A cos B − cos A sin B, then add these two equations together, the cos A sin B terms cancel, leaving 2 sin A cos B = sin(A+B) + sin(A−B). Dividing both sides by 2 gives the first product-to-sum identity. The other three are derived similarly by adding or subtracting the appropriate angle formulas.

When do you use product to sum identities vs. sum to product identities?

Use product to sum identities when you start with a product like sin A cos B and need to convert it into a sum. Use sum to product identities when you start with a sum like sin A + sin B and need to factor it into a product. In calculus, product to sum is especially helpful for integrating products of trig functions, while sum to product is often used to solve trig equations.

Do product to sum identities work for any angles?

Yes, these identities hold for all real values of A and B, whether the angles are given in degrees or radians. They are algebraic consequences of the addition formulas, which themselves are valid for all angles.

Product to Sum Identities vs. Sum to Product Identities

	Product to Sum Identities	Sum to Product Identities
Direction	Converts a product of trig functions into a sum/difference	Converts a sum/difference of trig functions into a product
Typical form (example)	sin A cos B = ½[sin(A+B) + sin(A−B)]	sin A + sin B = 2 sin[(A+B)/2] cos[(A−B)/2]
Common use in calculus	Integrating products like ∫ sin(mx) cos(nx) dx	Simplifying sums before solving equations
Number of identities	Four (sin·cos, cos·sin, cos·cos, sin·sin)	Four (sin±sin, cos±cos)

Why It Matters

You encounter product to sum identities frequently in precalculus and calculus, particularly when integrating products of trigonometric functions — without them, integrals like ∫ sin(3x)cos(5x) dx would be very difficult to evaluate. They also appear in physics and engineering, for instance in analyzing wave interference, where multiplying two wave signals and rewriting them as sums reveals beat frequencies. Mastering these identities strengthens your ability to manipulate and simplify a wide range of trigonometric expressions.

Common Mistakes

Mistake: Confusing the sign in the sin·sin identity. Students often write sin A sin B = ½[cos(A−B) + cos(A+B)] instead of the correct minus sign.

Correction: The correct identity is sin A sin B = ½[cos(A−B) − cos(A+B)]. A helpful way to remember: the sin·sin identity is the only one with a minus sign between the two terms on the right, and it produces cosines, not sines.

Mistake: Forgetting the factor of ½ in front of the sum.

Correction: Every product to sum identity has a ½ multiplier. This comes from dividing both sides by 2 during the derivation. Always include it, or your result will be off by a factor of 2.

Related Terms

Sum to Product Identities — The reverse process — converting sums into products
Trig Identities — The broader family of trigonometric identities
Sine — One of the trig functions involved in these identities
Cosine — The other trig function involved in these identities
Product — The multiplication operation being converted
Sum — The addition operation that results from the conversion
Angle Sum Identities — The addition formulas from which these identities are derived