Simplify $G(t) = \sum_\limits{n=1}^{\infty} (\cos\frac{n\pi}{2}-1) \cos(\frac{nt\pi}{3}) $

Question

Simplify $G(t) = \sum_\limits{n=1}^{\infty} (\cos\frac{n\pi}{2}-1) \cos(\frac{n\pi t}{3}) $

I am unsure how to simplify this in the best form, anyone have any ideas?

Answer 1

$$ (\cos\frac{n\pi}{2}-1) \cos(\frac{nt\pi}{3}) = \\ \frac{1}{2} \cos\left( n \left(\frac{\pi t}{3} - \frac{\pi}{2}\right)\right)+\frac{1}{2} \cos\left( n \left(\frac{\pi t}{3} + \frac{\pi}{2}\right)\right) -\cos \left( \frac{n \pi t}{3} \right) $$

So all terms are cosines.

Answer 2

Here is an alternate representation using the fact that

\begin{equation} \cos\left(\dfrac{n\pi}{2}\right)-1=\begin{cases} -1&\text{ for }n=4k-3,\,n=4k-1\\ -2&\text{ for }n=4k-2\\ \phantom{-}0&\text{ for }n=4k\\ \end{cases} \end{equation}

\begin{eqnarray*} G(t)&=&-\sum_{k=1}^{\infty}\left[\cos\left(\frac{4k-3}{3}\pi t\right) +2\cos\left(\frac{4k-2}{3}\pi t\right) +\left.\cos\left(\frac{4k-1}{3}\pi t\right)\right]\right. \end{eqnarray*}