Consider the following question
Following the hint, I got $$ \Phi (r, \theta) = \begin{cases} a_n r^n P_n(\cos \theta) \quad r>1 \ b_n r^{-(n+1)} P_n (\cos \theta) \quad r>1. \end{cases} $$ I tried to apply the initial condition and I got $$ \sum_{n=1}^{\infty} P_n (\cos \theta) ((-n-1)b_n -n a_n)= V \cos \theta^2 $$
However, I am not sure what to do now. I know that I can integrate both sides by
$$ \int P_k (\cos \theta) \cdot \big[\quad\big] \; d\theta, $$ but integrating
$$ \int P_k (\cos \theta) \cos ^2 \theta \; d \theta $$ has not been covered in my course.
Could someone clarify what is the intended way for this problem?