Product-to-Sum and Sum-to-Product Formulas | NIMCET

Quick answer: Products of sines and cosines can be written as sums, for example $2\sin A\cos B = \sin(A + B) + \sin(A - B)$. Sums can be written as products, for example $\sin C + \sin D = 2\sin\frac{C + D}{2}\cos\frac{C - D}{2}$. These transformation formulas make products easy to integrate or sum, and turn sums into products that can be cancelled or set to zero.

Both sets of formulas come from adding and subtracting the compound angle formulas. Knowing that link means you only need to remember the method, not eight separate results.

Product to sum

Add $\sin(A + B) = \sin A\cos B + \cos A\sin B$ and $\sin(A - B) = \sin A\cos B - \cos A\sin B$:

$2\sin A\cos B = \sin(A + B) + \sin(A - B)$

The other three come the same way:

Watch the last one: for $2\sin A\sin B$, the difference angle comes first.

Sum to product

Put $A + B = C$ and $A - B = D$, so $A = \frac{C + D}{2}$ and $B = \frac{C - D}{2}$:

The minus sign in the last row is the one students most often forget. An equivalent form is $\cos C - \cos D = 2\sin\frac{C + D}{2}\sin\frac{D - C}{2}$.

Worked examples

Example 1: simplifying a ratio

Simplify $\frac{\sin 5x + \sin 3x}{\cos 5x + \cos 3x}$.

Step 1: Numerator: $2\sin 4x\cos x$.

Step 2: Denominator: $2\cos 4x\cos x$.

Step 3: Cancel $2\cos x$ (assuming $\cos x \ne 0$):

$\frac{\sin 5x + \sin 3x}{\cos 5x + \cos 3x} = \tan 4x$

Example 2: cos 20° cos 40° cos 80°

Step 1: Multiply and divide by $2\sin 20^\circ$, and use $2\sin\theta\cos\theta = \sin 2\theta$ repeatedly:

$\frac{2\sin 20^\circ\cos 20^\circ\cos 40^\circ\cos 80^\circ}{2\sin 20^\circ} = \frac{\sin 40^\circ\cos 40^\circ\cos 80^\circ}{2\sin 20^\circ}$

Step 2: Repeat twice more:

$= \frac{\sin 80^\circ\cos 80^\circ}{4\sin 20^\circ} = \frac{\sin 160^\circ}{8\sin 20^\circ}$

Step 3: $\sin 160^\circ = \sin 20^\circ$, so the product is $\frac{1}{8}$.

The general pattern: $\cos\theta\cos 2\theta\cos 4\theta \cdots \cos 2^{n-1}\theta = \frac{\sin 2^n\theta}{2^n\sin\theta}$.

Example 3: the sin 60° − θ identity

Show that $\sin\theta\sin(60^\circ - \theta)\sin(60^\circ + \theta) = \frac{1}{4}\sin 3\theta$.

Step 1: Product to sum on the last two factors:

$\sin(60^\circ - \theta)\sin(60^\circ + \theta) = \sin^2 60^\circ - \sin^2\theta = \frac{3}{4} - \sin^2\theta$

Step 2: Multiply by $\sin\theta$:

$\sin\theta\left(\frac{3}{4} - \sin^2\theta\right) = \frac{3\sin\theta - 4\sin^3\theta}{4} = \frac{\sin 3\theta}{4}$

The cosine version, $\cos\theta\cos(60^\circ - \theta)\cos(60^\circ + \theta) = \frac{1}{4}\cos 3\theta$, works the same way. With $\theta = 20^\circ$ it gives $\cos 20^\circ\cos 40^\circ\cos 80^\circ = \frac{1}{4}\cos 60^\circ = \frac{1}{8}$ — a quick check on Example 2.

Example 4: solving with sum to product

Solve $\sin x + \sin 3x = 0$.

Step 1: $2\sin 2x\cos x = 0$.

Step 2: Either $\sin 2x = 0$, so $x = \frac{n\pi}{2}$, or $\cos x = 0$, so $x = (2n + 1)\frac{\pi}{2}$. The second set is contained in the first, so the solution is $x = \frac{n\pi}{2}$, $n \in \mathbb{Z}$.

Summary table

Exam tips

• Pair terms whose angles add or subtract to something useful (often a common factor appears).
• For a product of cosines whose angles double, multiply by $2\sin(\text{smallest angle})$.
• Recognise $20^\circ$, $40^\circ$, $80^\circ$ as $\theta$, $60^\circ - \theta$, $60^\circ + \theta$ with $\theta = 20^\circ$.
• When solving equations, change sums into products so that each factor can be set to zero.

Related: compound angle formulas and the NIMCET Trigonometry chapter.

Common mistakes

• Forgetting the negative sign in $\cos C - \cos D$.
• Putting the angles in the wrong order in $2\sin A\sin B = \cos(A - B) - \cos(A + B)$.
• Dividing by a factor such as $\cos x$ when solving an equation, which loses solutions.
• Forgetting the factor 2 on the product side of every formula.

Frequently Asked Questions

What is the formula for sin C + sin D?

sin C + sin D = 2 sin((C + D)/2) cos((C − D)/2).

What is 2 sin A sin B equal to?

2 sin A sin B = cos(A − B) − cos(A + B).

What is the value of cos 20° cos 40° cos 80°?

It is 1/8. Multiply and divide by 2 sin 20° and use the double angle formula three times.

Why are these called transformation formulas?

They transform a product of trigonometric functions into a sum, or a sum into a product, which makes simplifying, solving and integrating easier.

What is cos C − cos D?

cos C − cos D = −2 sin((C + D)/2) sin((C − D)/2).