Cosine to Power of Odd Integer/Mistake

Source Work

 * 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 $\cos^n \phi = \dfrac 1 {2^{n-1}} \left\{{\cos n \phi + n \cos \left({n - 2}\right) \phi + \dfrac {n \left({n- 1}\right)} 2 \cos \left({n - 4}\right) \phi + \cdots + R_n}\right\}$


 * where $R_n = \begin{cases} \cos \phi & \text{if $n$ is odd} \\ \dfrac {n!}{\left[{\left({n/2}\right)!}\right]^2} & \text {if $n$ is even.}\end{cases}$

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