<s>[INST] Discrete-time sinusoids with same frequency I've read that sine waves of the form $x_n = \sin(w_{0}n)$, with frequencies $w_{0}$ and $w_{0} + 2\pi$, are indistinguishable from each other when considering discrete time.
The book gives $\cos\big(\frac{\pi}{4}n\big)$ as an example. Adding $2\pi$ to the frequency, we get $\cos\big(\frac{9\pi}{4}n\big)$ which is indeed the same wave. However, the text's answer is $\cos\big(\frac{7\pi}{4}n\big)$ which is again the same wave I plot below. Which calculations lead to this answer?

Thanks in advance!
 [/INST] For $n$ an integer, 
$$\cos\frac{7\pi n}{4}= \cos\frac{9\pi n}{4} $$
because for $n$ even both are simply zero, and for $n$ odd, $\frac{7\pi n}{4}$ is  $-\left( \frac{9\pi n}{4} \right) + 2\pi$ but
$$
\cos\left[-\left( \frac{9\pi n}{4} \right) + 2\pi \right]= \cos[-\left( \frac{9\pi n}{4} \right)] = \cos\left[+\left( \frac{9\pi n}{4} \right)\right] $$
because cosine is an even function of its argument.
So both answers give the same graph.
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