Fourier transform of ${\Pi}_{a}(t)\cos(2{\pi}f_{0}t)$
Your last line is wrong. Note that you have

$$H(f)\star\delta(f-f_0)=H(f-f_0)\tag{1}$$

for any $H(f)$.

So your last line should be

$$\begin{align*}
&amp;a\,\text{sinc}(2af)\star\delta(f-f_{0})+a\text{ sinc}(2af)\star\delta(f+f_{0})=\\&amp;a\,\text{sinc}(2a(f-f_0))+a\,\text{sinc}(2a(f+f_0))\tag{2}
\end{align*}$$

You basically get copies of your original spectrum centered at $f_0$ and $-f_0$.