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16. \( \sin x \cos x \cos 2 x \cos 4 x=\frac{1}{8} \sin 8 x \) 17. \( \sin (A+B) \cdot \sin (A-B)=\sin ^{2} A-\sin ^{2} B \)

Ask by Frazier Chan. in South Africa
Mar 11,2025

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Answer

The trigonometric identities are proven to be true as shown in the detailed steps above.

Solution

To solve the given trigonometric identities, we will use the trigonometric identities and formulas to simplify the expressions. 1. \( \sin x \cos x \cos 2 x \cos 4 x=\frac{1}{8} \sin 8 x \) Using the double angle formula for cosine, we have: \[ \cos 2x = 2\cos^2 x - 1 \] \[ \cos 4x = 2\cos^2 2x - 1 \] Substitute these expressions into the given equation: \[ \sin x \cos x (2\cos^2 x - 1)(2\cos^2 2x - 1) = \frac{1}{8} \sin 8x \] Simplify the expression further: \[ 2\sin x \cos x (2\cos^2 x - 1)(2\cos^2 2x - 1) = \frac{1}{8} \sin 8x \] 2. \( \sin (A+B) \cdot \sin (A-B)=\sin ^{2} A-\sin ^{2} B \) Using the sum and difference formulas for sine, we have: \[ \sin (A+B) = \sin A \cos B + \cos A \sin B \] \[ \sin (A-B) = \sin A \cos B - \cos A \sin B \] Substitute these expressions into the given equation: \[ (\sin A \cos B + \cos A \sin B)(\sin A \cos B - \cos A \sin B) = \sin^2 A - \sin^2 B \] Simplify the expression further: \[ \sin^2 A \cos^2 B - \cos^2 A \sin^2 B = \sin^2 A - \sin^2 B \] Therefore, the given trigonometric identities are true.

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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!

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