Difference between single sided and double sided amplitude spectrum
For baseband real signals the frequency spectrum is conjugate symmetric; i.e., $$X(f) = X(-f)^*$$ This translates to an even magnitude spectrum; i.e., $$|X(f)| = |X(-f)|$$ and an odd phase spectrum.

Because of the symmetry it's also called as the double side band spectrum in commnnications terminology. For the real basband signal, a single side band is also sufficient to represent it. THere are different approaches to obtain the single side band as the uppersideband and the lower side band.

One method of obtaining the upper side band is to convert the signal into the analytic signal which is $x_+(t) = x(t) + j \hat{x}(t)$ where $\hat{x}(t)$ is the Hilbert transform of $x(t)$. The resulting spectrum is:
$$ X_+(f) = 
\begin{cases}
2 X(f) ~~, &amp;\text{ for} ~~ f &gt; 0 \\
0 ~~, &amp;\text{ for} ~~ f &lt; 0 \\
\end{cases}
$$

Therefore for the given signal $x(t) = \cos(2\pi 20000 t)$ whose spectrum is $$X(f) = 0.5 \delta(f-20000) + 0.5 \delta(f+20000)$$ the upper side band will be given by $$X_+(f) = \delta(f-20000)$$

Note that you can adjust the amplitude of single side band if you wish to do so.