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Sum to Product Identities — Formulas, Examples & Table

Sum to Product Identities

Trig identities which show how to rewrite sums and differences of sines or cosines as products.

 

Sum to Product Identities

sin x + sin y = 2 sin((x+y)/2) cos((x-y)/2)

sin x minus sin y equals 2 cos((x+y)/2) times sin((x-y)/2)

cos x + cos y = 2cos((x+y)/2)cos((x-y)/2)

cos x minus cos y equals negative 2 times sin((x+y)/2) times sin((x-y)/2)

 

See also

Product to sum identities

Key Formula

sin⁡A+sin⁡B=2sin⁡ ⁣(A+B2)cos⁡ ⁣(A−B2)\sin A + \sin B = 2 \sin\!\left(\frac{A+B}{2}\right)\cos\!\left(\frac{A-B}{2}\right) sin⁡A−sin⁡B=2cos⁡ ⁣(A+B2)sin⁡ ⁣(A−B2)\sin A - \sin B = 2 \cos\!\left(\frac{A+B}{2}\right)\sin\!\left(\frac{A-B}{2}\right) cos⁡A+cos⁡B=2cos⁡ ⁣(A+B2)cos⁡ ⁣(A−B2)\cos A + \cos B = 2 \cos\!\left(\frac{A+B}{2}\right)\cos\!\left(\frac{A-B}{2}\right) cos⁡A−cos⁡B=−2sin⁡ ⁣(A+B2)sin⁡ ⁣(A−B2)\cos A - \cos B = -2 \sin\!\left(\frac{A+B}{2}\right)\sin\!\left(\frac{A-B}{2}\right)
Where:
  • AA = First angle (in radians or degrees)
  • BB = Second angle (in radians or degrees)

Worked Example

Problem: Rewrite sin 75° + sin 15° as a product of trigonometric functions and evaluate it.
Step 1: Identify the identity to use. Since we have a sum of sines, apply the formula: sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2).
sin⁡75°+sin⁡15°=2sin⁡ ⁣(75°+15°2)cos⁡ ⁣(75°−15°2)\sin 75° + \sin 15° = 2 \sin\!\left(\frac{75° + 15°}{2}\right)\cos\!\left(\frac{75° - 15°}{2}\right)
Step 2: Compute the half-sum and half-difference of the angles.
75°+15°2=45°,75°−15°2=30°\frac{75° + 15°}{2} = 45°, \qquad \frac{75° - 15°}{2} = 30°
Step 3: Substitute the values into the formula.
2sin⁡45°cos⁡30°=2⋅22⋅322 \sin 45° \cos 30° = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}
Step 4: Simplify the product.
2⋅64=622 \cdot \frac{\sqrt{6}}{4} = \frac{\sqrt{6}}{2}
Answer: sin 75° + sin 15° = √6 / 2 ≈ 1.2247

Another Example

Problem: Express cos 5x − cos 3x as a product.
Step 1: Use the cosine difference identity: cos A − cos B = −2 sin((A+B)/2) sin((A−B)/2).
cos⁡5x−cos⁡3x=−2sin⁡ ⁣(5x+3x2)sin⁡ ⁣(5x−3x2)\cos 5x - \cos 3x = -2\sin\!\left(\frac{5x+3x}{2}\right)\sin\!\left(\frac{5x-3x}{2}\right)
Step 2: Simplify the arguments.
=−2sin⁡(4x)sin⁡(x)= -2\sin(4x)\sin(x)
Answer: cos 5x − cos 3x = −2 sin(4x) sin(x)

Frequently Asked Questions

How do you derive the sum to product identities?
Start with the product to sum identities (also called prosthaphaeresis formulas) and solve them in reverse. For example, begin with the known identities sin(X + Y) + sin(X − Y) = 2 sin X cos Y. Then set A = X + Y and B = X − Y, which gives X = (A + B)/2 and Y = (A − B)/2. Substituting back yields sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2). The other three formulas are derived the same way.
When should I use sum to product versus product to sum?
Use sum to product when you need to factor a sum or difference of trig functions—this is common when solving equations (setting a factor equal to zero) or simplifying expressions. Use product to sum when you need to expand a product into additive terms, which often arises in integration or signal processing.

Sum to Product Identities vs. Product to Sum Identities

These two sets of identities are inverses of each other. Sum to product identities convert additions like sin A + sin B into a product of trig functions. Product to sum identities do the opposite: they expand a product like sin A cos B into a sum. You choose which set to use based on whether your goal is to factor (sum → product) or expand (product → sum).

Why It Matters

Sum to product identities let you factor trigonometric expressions, which is essential for solving equations of the form sin A + sin B = 0—you convert the sum to a product and then set each factor to zero. They also simplify certain integrals in calculus and appear in physics when analyzing wave interference, where two waves combining can be rewritten as a single modulated wave using these formulas.

Common Mistakes

Mistake: Forgetting the negative sign in the cos A − cos B identity.
Correction: The formula is cos A − cos B = −2 sin((A+B)/2) sin((A−B)/2). The leading factor is −2, not +2. A good habit is to verify with specific values (e.g., A = 90°, B = 0°) to confirm the sign.
Mistake: Swapping sine and cosine in the result for sin A − sin B.
Correction: For sin A − sin B, the result is 2 cos((A+B)/2) sin((A−B)/2)—note that cos comes first and sin comes second, which is the reverse of the sin A + sin B formula. Mixing these up changes the answer entirely.

Related Terms

  • Product to Sum Identities — Inverse identities that expand products into sums
  • Trig Identities — The broader family of trigonometric formulas
  • Sine — One of the core functions in these identities
  • Cosine — The other core function in these identities
  • Sum — The additive operation being converted
  • Product — The multiplicative form produced by these identities
  • Difference — Subtraction cases also covered by these identities