Autocorrelation of discrete signal $\sin{2\pi fn}$ Determine the autocorrelation $r_{xx}[m]$ of the discrete signal 
  
  $$x[n] = (\sin2\pi fn).$$
  
  where $n$ and $m$ are integers. 


Using the definition I get

$$\begin{align}
r_{xx}[m] &amp;= \sum_{n=-\infty}^{\infty}x[n]x[n-m] \\
&amp;= \sum_{n=-\infty}^{\infty}\sin(2\pi fn) \sin\big(2\pi f(n-m)\big) \\
&amp;= \sum_{n=-\infty}^{\infty}\sin(2\pi fn) \sin\big(2\pi fn - 2\pi fm\big) \\
\end{align}$$

but I can't seem to figure it out from here. I've tried using different trigonometric identities without result. I'm guessing it's something simple I'm missing.
well, there's an old trig identity that you learned pre-calculus:

$$ \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) $$

or, alternatively

$$ \sin(\alpha) \sin(\beta) = \tfrac12 \big(\cos(\alpha-\beta) - \cos(\alpha+\beta)\big) $$

and you will need to use one of those.

but you have a bigger problem because $x[n] = \sin(2\pi f n)$ is a finite power, infinite energy signal.  the definition for autocorrelation is different, i believe for power signals than for energy signals.

the equation for crosscorrelation for power signals is, i believe,

$$ r_{xy}[m] \ = \ \lim_{N \to \infty} \ \ \tfrac{1}{2N+1} \sum_{n=-N}^{N}x[n] \overline{y[n-m]} $$

for autocorrelation of a purely real signal

$$ r_{xx}[m] \ = \ \lim_{N \to \infty} \ \ \tfrac{1}{2N+1} \sum_{n=-N}^{N}x[n] x[n-m] $$