Quadruple Angle Formulas/Sine/Mistake

Source Work

 * Chapter $1$: Complex Numbers
 * Supplementary Problems: $93 \ \text{(a)}$

This mistake can be seen in the 1981 printing of the second edition (1974) as published by Schaum: ISBN 0-070-84382-1

Mistake

 * Prove that:
 * $\dfrac {\sin 4 \theta} {\sin \theta} = 8 \cos^3 \theta - 4 = 2 \cos 3 \theta + 6 \cos \theta - 4$

Correction
The correct expression is:
 * $\dfrac {\sin 4 \theta} {\sin \theta} = 8 \cos^3 \theta - 4 \cos \theta = 2 \cos 3 \theta + 2 \cos \theta$

as demonstrated in Quadruple Angle Formulas for Sine: Corollary 1 and Quadruple Angle Formulas for Sine: Corollary 2

The mistake in the second expression probably arose from taking the incorrect result of the first expression:
 * $\dfrac {\sin 4 \theta} {\sin \theta} = 8 \cos^3 \theta - 4$

and substituting for $\cos^3 \theta$ from Power Reduction Formulas/Cosine Cubed.