According to the Balanis's book (C.A. Balanis, "Antenna theory- Analysis and design", John Wiley and Sons Inc), the sinusoidal form is experimentally verified fact.
As long as I know, it seems there is no mathematical proof.
The following statements are taken from Balanis' book:
4.5.1 Current Distribution
For a very thin dipole (ideally zero diameter), the current distribution can be written,
to a good approximation, as
\begin{equation*}
\mathbf{I}_l(x'=0,y'=0,z')=
\begin{cases}
\hat{\mathbf{a}}_zI_0\text{sin}\left[k\left(\frac{l}{2}-z'\right)\right],\;\;\;(0\leq z'\leq l/2)\\
\hat{\mathbf{a}}_zI_0\text{sin}\left[k\left(\frac{l}{2}+z'\right)\right],\;\;\;(-l/2 \leq z'\leq 0)
\end{cases} \text{ (4-56)}
\end{equation*}
This distribution assumes that the antenna is center-fed and current vanishes at
the end points ($z'=\pm l/2)$. Experimentally, it has been verified that the current in a
center-fed wire antenna has sinusoidal form with nulls at the end points.
For $l=\lambda/2$ and $\lambda/2<l<\lambda$ the current distribution (4-56) is shown plotted
in Figures 1.16(b) and 1.12(c), respectively. The geometry of the antenna is that shown in Figure 4.5
