Sum-to-Product Formulas

Why Convert a Sum into a Product?

A sum of two trigonometric terms can sometimes be easier to understand or simplify when written as a product. The sum-to-product formulas make this conversion systematic: they use the average of the two angles and half their difference.

For example, \(\sin(7x)+\sin(3x)\) has two sine terms. Its sum-to-product form is a single product involving \(\sin(5x)\) and \(\cos(2x)\). The formulas work for angles that are numbers, such as \(75^\circ\), or expressions involving a variable, such as \(7x\).

Definition: For angles \(u\) and \(v\), their average is \(\frac{u+v}{2}\), and half their difference, in the order \(u-v\), is \(\frac{u-v}{2}\). Sum-to-product formulas express sums or differences of trigonometric functions of \(u\) and \(v\) using these two angles.

Use degrees throughout a calculation or radians throughout a calculation; do not mix the two. The formulas apply to real angles in either unit, provided the angle units are consistent. The signs and the order of \(u\) and \(v\) matter, particularly for differences.

The Four Formulas

The four identities cover a sum and a difference of sines, and a sum and a difference of cosines:

\sin u+\sin v = 2\sin\left(\frac{u+v}{2}\right) \cos\left(\frac{u-v}{2}\right)

\sin u-\sin v = 2\cos\left(\frac{u+v}{2}\right) \sin\left(\frac{u-v}{2}\right)

\cos u+\cos v = 2\cos\left(\frac{u+v}{2}\right) \cos\left(\frac{u-v}{2}\right)

\cos u-\cos v = -2\sin\left(\frac{u+v}{2}\right) \sin\left(\frac{u-v}{2}\right)

Notice that the difference formulas use \((u-v)/2\), not \((v-u)/2\), when the left side is written in the order \(u-v\). In the cosine difference formula, there is also a negative sign in front. Keeping the same order of angles throughout is a reliable way to avoid sign errors.

Why the Formulas Work

The identities follow from the angle-addition formulas. The essential idea is to express both angles as the average plus or minus half the difference. Set

m=\frac{u+v}{2} \qquad\text{and}\qquad d=\frac{u-v}{2}.

Then \(m+d=u\) and \(m-d=v\). Apply the angle-addition formulas to \(\sin(m+d)\) and \(\sin(m-d)\):

\begin{aligned} \sin u &=\sin(m+d) =\sin m\cos d+\cos m\sin d,\\ \sin v &=\sin(m-d) =\sin m\cos d-\cos m\sin d. \end{aligned}

Adding these equations cancels the terms \(\cos m\sin d\) and \(-\cos m\sin d\). Subtracting the second equation from the first cancels the two terms \(\sin m\cos d\). Thus

\begin{aligned} \sin u+\sin v&=2\sin m\cos d,\\ \sin u-\sin v&=2\cos m\sin d. \end{aligned}

Substituting the definitions of \(m\) and \(d\) gives the two sine formulas. For cosine, the angle-addition formulas give

\begin{aligned} \cos u &=\cos(m+d) =\cos m\cos d-\sin m\sin d,\\ \cos v &=\cos(m-d) =\cos m\cos d+\sin m\sin d. \end{aligned}

Adding cancels the sine-product terms; subtracting gives \(-2\sin m\sin d\). Therefore

\begin{aligned} \cos u+\cos v&=2\cos m\cos d,\\ \cos u-\cos v&=-2\sin m\sin d. \end{aligned}

Replacing \(m\) and \(d\) by \((u+v)/2\) and \((u-v)/2\) proves both cosine formulas as well. This derivation also explains the negative sign in the cosine difference identity: it comes directly from subtracting \(\cos(m-d)\) from \(\cos(m+d)\).

Worked Examples

Worked Example: Rewrite a Sum of Sines

Rewrite \(\sin(7x)+\sin(3x)\) as a product.

Match the expression to \(\sin u+\sin v\) by taking \(u=7x\) and \(v=3x\). Compute the average angle and half-difference:

\frac{u+v}{2} =\frac{7x+3x}{2} =5x, \qquad \frac{u-v}{2} =\frac{7x-3x}{2} =2x.

Apply the sine-sum formula:

\sin(7x)+\sin(3x) = 2\sin(5x)\cos(2x).

Check by using the product-to-sum identity \(2\sin A\cos B=\sin(A+B)+\sin(A-B)\):

\begin{aligned} 2\sin(5x)\cos(2x) &=\sin(5x+2x)+\sin(5x-2x)\\ &=\sin(7x)+\sin(3x). \end{aligned}

The product expands back to the original expression.

Worked Example: Evaluate a Sum Using Exact Values

Find the exact value of \(\sin 75^\circ+\sin 15^\circ\).

Here \(u=75^\circ\) and \(v=15^\circ\). The average angle and half-difference are

\frac{u+v}{2} =\frac{75^\circ+15^\circ}{2} =45^\circ, \qquad \frac{u-v}{2} =\frac{75^\circ-15^\circ}{2} =30^\circ.

The sine-sum formula gives

\begin{aligned} \sin 75^\circ+\sin 15^\circ &=2\sin 45^\circ\cos 30^\circ\\ &=2\left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right)\\ &=\frac{\sqrt{6}}{2}. \end{aligned}

Check using the exact values of the original terms:

\sin 75^\circ+\sin 15^\circ = \frac{\sqrt{6}+\sqrt{2}}{4} + \frac{\sqrt{6}-\sqrt{2}}{4} = \frac{2\sqrt{6}}{4} = \frac{\sqrt{6}}{2}.

The exact value agrees with the result from the formula.

Worked Example: Simplify a Difference of Cosines

Rewrite \(\cos(5x)-\cos(x)\) as a product.

Set \(u=5x\) and \(v=x\). Then

\frac{u+v}{2} =\frac{5x+x}{2} =3x, \qquad \frac{u-v}{2} =\frac{5x-x}{2} =2x.

Use the cosine-difference formula, including its negative sign:

\cos(5x)-\cos(x) = -2\sin(3x)\sin(2x).

Check by expanding the product with \(2\sin A\sin B=\cos(A-B)-\cos(A+B)\):

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

The order and sign match the original difference.

Worked Example: Simplify Shifted Sines

Simplify \(\sin(x+\frac{\pi}{6})+\sin(x-\frac{\pi}{6})\).

Choose \(u=x+\frac{\pi}{6}\) and \(v=x-\frac{\pi}{6}\). Their average and half-difference are

\begin{aligned} \frac{u+v}{2} &=\frac{(x+\frac{\pi}{6})+(x-\frac{\pi}{6})}{2} =x,\\ \frac{u-v}{2} &=\frac{(x+\frac{\pi}{6})-(x-\frac{\pi}{6})}{2} =\frac{\pi}{6}. \end{aligned}

Thus

\begin{aligned} \sin\left(x+\frac{\pi}{6}\right) +\sin\left(x-\frac{\pi}{6}\right) &=2\sin x\cos\left(\frac{\pi}{6}\right)\\ &=2\sin x\left(\frac{\sqrt{3}}{2}\right)\\ &=\sqrt{3}\sin x. \end{aligned}

To check, expand each sine using the angle-addition formula:

\begin{aligned} \sin\left(x+\frac{\pi}{6}\right) +\sin\left(x-\frac{\pi}{6}\right) &=\left(\frac{\sqrt{3}}{2}\sin x+\frac12\cos x\right) +\left(\frac{\sqrt{3}}{2}\sin x-\frac12\cos x\right)\\ &=\sqrt{3}\sin x. \end{aligned}

The cosine terms cancel, confirming the simplified result.

A Reliable Way to Choose the Formula

First identify whether the two terms involve sine or cosine, and whether they are added or subtracted. Then keep the original order when assigning \(u\) and \(v\). Finally calculate the average and half-difference before substituting into the matching formula.

Identify the pattern.
Match the expression to a sine sum, sine difference, cosine sum, or cosine difference.

Label the angles in order.
Write the first angle as \(u\) and the second as \(v\), preserving the order of subtraction if there is a difference.

Calculate both combinations.
Find \((u+v)/2\) and \((u-v)/2\) carefully, including parentheses when the angles contain several terms.

Substitute and check.
Use the matching identity, then expand the product with a product-to-sum identity or the angle-addition formulas to verify it.

A Common Sign Error

A frequent mistake is to omit the negative sign in the cosine-difference formula. For example, a mistaken substitution for \(\cos u-\cos v\) might give \(2\sin((u+v)/2)\sin((u-v)/2)\), with no minus sign.

Test the proposed formula with \(u=60^\circ\) and \(v=0^\circ\). The original expression is

\cos 60^\circ-\cos 0^\circ =\frac12-1 =-\frac12.

The incorrect formula gives

2\sin 30^\circ\sin 30^\circ =2\left(\frac12\right)\left(\frac12\right) =\frac12,

which has the wrong sign. The correct formula gives

-2\sin 30^\circ\sin 30^\circ=-\frac12,

as required. A quick numerical check can expose a sign mistake, but the general correction is to use the cosine-difference identity with its leading negative sign.

Why These Formulas Are Useful

Sum-to-product formulas can reveal cancellations, reduce a pair of trigonometric terms to one product, or make exact evaluation more direct. They are especially convenient when the angles are symmetric around a simple average: in the shifted-sine example, the terms were centered at \(x\), and the half-difference was the fixed angle \(\pi/6\).

They are not a command to transform every sum. A product may or may not be simpler for a particular task. When you do use one, the essential checks are that you have selected the right sum-or-difference formula, computed both angles correctly, and preserved the sign. These habits make the identities dependable tools for rewriting expressions.

Check Your Understanding

For each question, keep track of the order of the two angles and show the average and half-difference where appropriate.

Rewrite \(\sin(9x)+\sin(5x)\) as a product.
Rewrite \(\sin(4x)-\sin(2x)\) as a product.
Rewrite \(\cos(8x)+\cos(2x)\) as a product.
Rewrite \(\cos(7x)-\cos(3x)\) as a product, including the correct sign.
Evaluate \(\sin 80^\circ+\sin 40^\circ\) exactly using a sum-to-product formula.

What You'll Learn

Why Convert a Sum into a Product?

The Four Formulas

Why the Formulas Work