Sum to Product: cos A + cos B
Sum-to-product identity.
Formula
Variables
How it works
This identity rewrites the sum of two cosine terms as a product of two cosines: one depending on the average angle and one depending on half the difference. It comes from the cosine addition and subtraction formulas, by adding \(\cos(A+B)\) and \(\cos(A-B)\) and then rearranging. The result is especially useful because it can turn a messy sum into a form that reveals symmetry, cancellation, or amplitude modulation.
When to use it
Use it when you need to simplify expressions like \(\cos 7x + \cos 3x\), factor trigonometric sums, or analyze interference patterns in physics and signal processing. It is also useful for solving equations where two cosine terms share related angles.
Worked example, step by step
Simplify \(\cos 80^\circ + \cos 20^\circ\). Use the identity \(\cos A + \cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2}\). Set \(A=80^\circ\), \(B=20^\circ\). \(\cos 80^\circ + \cos 20^\circ = 2\cos\frac{80^\circ+20^\circ}{2}\cos\frac{80^\circ-20^\circ}{2}\) \(= 2\cos 50^\circ\cos 30^\circ\) \(= 2(0.6428)(0.8660)\) \(\approx 1.1132\) So, \(\cos 80^\circ + \cos 20^\circ \approx 1.1132\).
Common mistakes to avoid
- Using \(\frac{A-B}{2}\) without keeping the order consistent, which can confuse the sign in other sum-to-product identities
- Forgetting the factor of 2 in front of the product
- Averaging the angles incorrectly, such as using \(\cos(A+B)\) instead of \(\cos\frac{A+B}{2}\)
Did You Know?
Fourier showed that any repeating signal is a sum of cosines and sines like these — the basis of MP3 and JPEG.
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More in Trigonometry
View allSine Ratio (SOH)
BasicRatio of the side opposite an angle to the hypotenuse.
Cosine Ratio (CAH)
BasicRatio of the side adjacent to an angle to the hypotenuse.
Tangent Ratio (TOA)
BasicRatio of the opposite side to the adjacent side.
Cosecant
IntermediateReciprocal of sine.
Secant
IntermediateReciprocal of cosine.
Cotangent
IntermediateReciprocal of tangent.