Why Convert a Product into a Sum?
A product such as \(\sin A\cos B\) can be awkward to integrate directly, especially when the angles \(A\) and \(B\) differ. A product-to-sum identity rewrites it as a sum or difference of individual sine or cosine terms. Those terms are often easier to handle because their antiderivatives are standard.
The identities are also connected to power-reduction formulas. Setting the two angles equal turns products such as \(\sin^2 x\) and \(\cos^2 x\) into expressions involving a constant and a double angle. The main goal here is to derive these identities carefully and understand what each one does.
The Three Identities
For any real angles \(A\) and \(B\), the product-to-sum formulas are:
The order of the angles matters in the last formula: it is written for \(\sin A\cos B\). For the reversed product, the corresponding identity is
These are identities, not integration rules that apply only in a particular setting. They hold for every pair of real angles. In an integral, however, their usefulness depends on whether the resulting terms are simpler to integrate than the original product.
Derivation from Angle-Addition Formulas
The cosine addition and subtraction formulas are
Subtracting the first equation from the second cancels the \(\cos A\cos B\) terms and leaves \(2\sin A\sin B\). Adding them cancels the sine products and leaves \(2\cos A\cos B\). The sine-product and cosine-product identities therefore follow directly.
\(\cos(A-B)-\cos(A+B)=2\sin A\sin B\). Dividing by \(2\) gives the sine-sine identity.
\(\cos(A-B)+\cos(A+B)=2\cos A\cos B\). Dividing by \(2\) gives the cosine-cosine identity.
The sine addition and subtraction formulas isolate the sine-cosine product, as shown below.
For the remaining identity, use the sine addition and subtraction formulas:
Adding these equations cancels the \(\cos A\sin B\) terms. The left side is the sum of the two sine terms, and the right side is \(2\sin A\cos B\). Dividing by \(2\) proves the identity.
Using the Identities When the Angles Match
When \(A=B\), the sine-sine identity gives a constant minus a cosine term, and the cosine-cosine identity gives a constant plus a cosine term. These are the power-reduction identities for the squares. When both factors have the same angle, the sine-cosine identity gives a single sine term with a doubled angle. This makes each product straightforward to integrate using basic antiderivatives.
Worked Example: Integrating a Sine-Squared Product
Rewrite and integrate \(\sin^2(3x)\). Set \(A=B=3x\) in the sine-sine identity:
Since \(\cos 0=1\), this becomes
Integrate each term:
To check, differentiate the result:
The derivative matches the integrand.
Worked Example: Integrating a Cosine-Squared Product
Rewrite and integrate \(\cos^2(2x)\). Set \(A=B=2x\) in the cosine-cosine identity:
Using \(\cos 0=1\), obtain
Now integrate:
Differentiate to verify:
The derivative recovers the original integrand.
Worked Example: Integrating a Sine-Cosine Product
Use the sine-cosine identity to integrate \(\sin(4x)\cos(4x)\). With \(A=B=4x\),
Since \(\sin 0=0\), integrate the resulting single sine term:
Check by differentiation:
Thus the antiderivative differentiates to the stated product.
Recognizing When a Product-to-Sum Formula Helps
Each formula changes a particular kind of product into a sum of one-family terms. The choice is determined by the two factors:
| Product | Result of the identity | Useful when |
|---|---|---|
| \(\sin A\sin B\) | A difference of cosines | Integrating the resulting cosine terms is more direct than handling the product. |
| \(\cos A\cos B\) | A sum of cosines | The individual cosine terms have simpler antiderivatives than the original product. |
| \(\sin A\cos B\) | A sum of sines | The sine terms are easier to integrate separately than the product. |
If the angles match, one of the resulting angles is zero. The corresponding cosine term is the constant \(1\), or the corresponding sine term is \(0\). That is why the formulas simplify squares and same-angle sine-cosine products so effectively.
If the angles do not match, the formulas instead produce terms involving the sum and difference of the angles. This can still be helpful: a product becomes a pair of individual trigonometric terms, each of which may be integrated separately. Before applying the identity, check that the resulting angle expressions are simpler for the integral at hand. A product-to-sum rewrite is not automatically an improvement if it creates terms that are no easier to work with.
A Common Sign Error
The sine-sine formula requires a difference of cosines in the order \(\cos(A-B)-\cos(A+B)\). Reversing the sign and adding the two cosine expressions gives the cosine-cosine product instead. For example, the incorrect claim
fails when \(A=B=\frac{\pi}{3}\). Its left side is
whereas its right side is
The values are different, so the claimed identity is false. The correction is to subtract the cosine formulas to isolate the sine product:
A reliable way to avoid this mistake is to write both angle-addition equations before adding or subtracting them. The terms that should cancel then determine the correct operation.
Check Your Understanding
Use the angle-addition formulas and the product-to-sum identities to answer these questions.
- Which operation on the two cosine angle-addition formulas gives the sine-sine product identity?
- Write \(\cos A\cos B\) as a sum of cosines.
- Use the sine-cosine identity to rewrite \(\sin(2x)\cos(2x)\).
- Why does setting \(A=B\) in the sine-sine identity introduce a constant term?
- What should you check before deciding that a product-to-sum rewrite will simplify an integral?
