Question

cos(n/8)cos(3n/8) periodic or not???

Answer 1

\[\cos(x\pm y)=\cos x\cos y\mp \sin x\sin y\]
If you were to add the two identities together, you'd have
\[\cos(x+y)+\cos(x-y)=2\cos x\cos y\]
which means you can rewrite the given expression as
\[\frac{\cos\left(\dfrac{n}{8}+\dfrac{3n}{8}\right)+\cos\left(\dfrac{n}{8}-\dfrac{3n}{8}\right)}{2}\]
Simplify:
\[\frac{\cos\dfrac{4n}{8}+\cos\left(-\dfrac{2n}{8}\right)}{2}\\
\frac{1}{2}\cos\dfrac{n}{2}+\frac{1}{2}\cos\dfrac{n}{4}\]
The period of a co/sine function of the form \(\sin bx\) or \(\cos bx\) is \(\dfrac{2\pi}{b}\). However, you have a sum of two period functions with different periods. The period of a sum of periodic functions is the least common multiple of the periods of each function.

Period of \(\cos\dfrac{n}{2}\) is \(\dfrac{2\pi}{1/2}=4\pi\). Period of \(\cos\dfrac{n}{4}\) is \(\dfrac{2\pi}{1/4}=8\pi\). The LCM is \(8\pi\).

http://www.wolframalpha.com/input/?i=period+of+cos%28n%2F8%29*cos%283n%2F8%29