Cosine to Power of Odd Integer/Mistake

Source Work

1964: Murray R. Spiegel: Theory and Problems of Complex Variables

Chapter $1$: Complex Numbers

Supplementary Problems: $130 \ \text{(a)}$

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 $\cos^n \phi = \dfrac 1 {2^{n - 1} } \set {\cos n \phi + n \cos \paren {n - 2} \phi + \dfrac {n \paren {n - 1} } 2 \cos \paren {n - 4} \phi + \cdots + R_n}$

where $R_n = \begin{cases} \cos \phi & \textit {if $n$ is odd} \\ \dfrac {n!} {\sqbrk {\paren {n / 2}!}^2} & \textit {if $n$ is even.}\end{cases}$

Correction

As demonstrated in Cosine to Power of Odd Integer, the last term in the odd expansion is not $\cos \phi$, it is $\dfrac {n!} {\paren {\frac {n + 1} 2}! \paren {\frac {n - 1} 2}!} \cos \phi$.

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $130 \ \text{(a)}$