16. \( \sin x \cos x \cos 2 x \cos 4 x=\frac{1}{8} \sin 8 x \) 17. \( \sin (A+B) \cdot \sin (A-B)=\sin ^{2} A-\sin ^{2} B \)
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The Deep Dive
The first equation \( \sin x \cos x \cos 2x \cos 4x = \frac{1}{8} \sin 8x \) can be derived using several trigonometric identities. This equation effectively relates the product of sine and cosine functions to a sine of a multiple angle. It's reminiscent of the transformations used in Fourier series, exploring how simple periodic functions can combine to create complex waveforms! The second equation \( \sin(A+B) \cdot \sin(A-B) = \sin^2 A - \sin^2 B \) is a beautiful application of the product-to-sum identities in trigonometry. It showcases how we can express products of sines as differences of squares, paving the way for further manipulations in calculus and algebra. This identity also serves as a stepping stone into deeper explorations in signal processing and harmonic analysis!