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Product-to-sum identities turn a product of sines and cosines into a sum, which is what makes Fourier analysis and beat frequencies tractable: sin A sin B = ½[cos(A − B) − cos(A + B)].
Sine × sine
sin A sin B = ½[cos(A − B) − cos(A + B)]
Cosine × cosine
cos A cos B = ½[cos(A − B) + cos(A + B)]
Sine × cosine
sin A cos B = ½[sin(A + B) + sin(A − B)]
The identity is exact, so ½[cos 30° − cos 70°] equals sin 50° × sin 20° ≈ 0.2620026 to full precision.
In signal processing: multiplying two tones produces sum and difference frequencies, which is the mathematics behind amplitude modulation and audible beats.