<s>[INST] Simplify $G(t) = \sum_\limits{n=1}^{\infty} (\cos\frac{n\pi}{2}-1) \cos(\frac{nt\pi}{3}) $ 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?
 [/INST] $$ (\cos\frac{n\pi}{2}-1) \cos(\frac{nt\pi}{3}) = \\ \frac{1}{2} \cos\left( n \left(\frac{\pi t}{3} - \frac{\pi}{2}\right)\right)+\frac{1}{2} \cos\left( n \left(\frac{\pi t}{3} + \frac{\pi}{2}\right)\right) -\cos \left( \frac{n \pi t}{3} \right) $$
So all terms are cosines. 
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