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

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.