= Solution
For a separated <harmonic function> $R(r)\Theta(\theta)$ in <polar coordinates>, single-valuedness forces integer angular frequencies. The radial factors for frequency $m\geq1$ are $r^m,r^{-m}$, while frequency zero has a constant and $\ln r$. Finiteness at the origin excludes $r^{-m}$ and $\ln r$.
The source's angular factor is $\cos\theta$ in every term, as printed, rather than $\cos(n\theta)$. The <power-law angular mode for the planar Poisson equation> gives
$$
\Delta[r^{n+2}\cos\theta]=[(n+2)^2-1]r^n\cos\theta=(n+1)(n+3)r^n\cos\theta.
$$
Hence, assuming the series permits termwise differentiation, the general finite, single-valued solution is
$$
\boxed{\phi=C+\sum_{m=1}^\infty r^m[A_m\cos(m\theta)+B_m\sin(m\theta)]+\cos\theta\sum_{n=0}^\infty\frac{\alpha_n r^{n+2}}{(n+1)(n+3)}.}
$$
The arbitrary harmonic series must itself converge to a finite function on the prescribed disk. If $\alpha_0\ne0$, the source has a direction-dependent limit at the origin; the displayed potential still extends finitely there, but the equation at that point is understood weakly rather than as a smooth classical source.
For the special source, a particular potential is $r^3\cos\theta/8$. Its boundary cosine term is canceled by $-a^2r\cos\theta/8$, and the boundary sine term is supplied by $(r/a)^2\sin2\theta$. The resulting <Dirichlet boundary conditions> has solution
$$
\boxed{\phi(r,\theta)=\frac{r(r^2-a^2)}8\cos\theta+\frac{r^2}{a^2}\sin2\theta.}
$$
It is finite at zero, has the required boundary values, and its <Laplacian> is $r\cos\theta$.
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