# Sum to Product Formulas — Trig Identities, Proof

The sum to product formula family converts a sum or difference of two sines (or two cosines) into a product of one sine and one cosine — four identities that turn sin⁡75°+sin⁡15° into a single product expression solvable in one step. This article gives the four formulas, the proof via sum-and-difference identities, three worked examples in degrees and radians, and the common mistake of mixing up the half-sum and half-difference angles.

## Four Formulas That Turn Stubborn Sums Into Easy Products

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

\[ \sin A + \sin B = 2 \sin\left(\dfrac{A+B}{2}\right) \cos\left(\dfrac{A-B}{2}\right) \]

\[ \sin A - \sin B = 2 \cos\left(\dfrac{A+B}{2}\right) \sin\left(\dfrac{A-B}{2}\right) \]

\[ \cos A + \cos B = 2 \cos\left(\dfrac{A+B}{2}\right) \cos\left(\dfrac{A-B}{2}\right) \]

\[ \cos A - \cos B = -2 \sin\left(\dfrac{A+B}{2}\right) \sin\left(\dfrac{A-B}{2}\right) \]

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_.**

> **Quick facts**  
> - **Hold for all real A, B** — no domain restrictions.
> - **Companion identities — product to sum:** the reverse direction is 2sin⁡Xcos⁡Y=sin⁡(X+Y)+sin⁡(X−Y), 2cos⁡Xcos⁡Y=cos⁡(X+Y)+cos⁡(X−Y), 2sin⁡Xsin⁡Y=cos⁡(X−Y)−cos⁡(X+Y).
> - **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.

## Double-Anchoring — Right Triangle and Unit Circle

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

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:

\[ \sin A + \sin B = 2 \sin X \cos Y \]

\[ \sin A - \sin B = \sin X \cos Y - \cos X \sin Y \]

Add the two equations:

\[ \sin A + \sin B = 2 \sin X \cos Y = 2 \sin\left(\dfrac{A+B}{2}\right) \cos\left(\dfrac{A-B}{2}\right) \]

**On the unit circle.** When A and B are close, (A−B)/2 is small, thus \(\cos((A−B)/2)\approx 1\), yielding the average of the two angles' sines, doubled.

## Three Worked Examples — Quick, Standard, Stretch

**Quick.** Evaluate \(\sin 75°+\sin 15°\) exactly.

Apply \(\sin A + \sin B = 2 \sin\left(\dfrac{A+B}{2}\right) \cos\left(\dfrac{A−B}{2}\right)\) with A=75° and 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: \(\sin 45° = \sqrt{2}/2\) and \(\cos 30° = \sqrt{3}/2\).

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

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

**Standard.** Evaluate \(\cos 75°−\cos 15°\) exactly.

Apply the correct identity:

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

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

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

**Stretch.** Express \(\sin x + \sin 3x + 
\sin 5x + \sin 7x\) as a product.

Using the sum-to-product identities leads to:

\[ 4 \sin 4x \cos 2x \cos x. \]

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

**Final answer:** \(\sin x + \sin 3x + 
\sin 5x + \sin 7x = 4 \sin 4x \cos 2x \cos x.\)

## 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.

### **Frequently Asked Questions**

- What are the sum to product formulas?  
  Four identities that rewrite \(\sin A ± \sin B\) and \(\cos A ± \cos B\) as products of one sine and one cosine.
- Why are they useful?  
  They convert sums into products, simplifying evaluations.
-  Is \(\sin A + \sin B = \sin(A + B)\)?  
   No — sin does not distribute over addition.
- When is sum to product preferred over expanding via angle addition?  
  When the two angles share special values.
- Are these formulas in the JEE syllabus?  
   Yes — Class 11 trig identities chapter.