Summation $\sum_{n=0}^{\infty} \frac{P_n(\cos \theta)}{n+1} = \log\frac{\sin(\theta/2) +1}{\sin(\theta/2)}$

Question

$P_n$ is legendres polynomial of $n$ degree terms. I tried to use the formula summation of $C_n P_n(x) = f(x)$ but not able to find $f(x)$. Also I tried using generating functions but that too didn't help.

Answer 1

Let $\cos \theta =c$, $$S=\sum_{n=0}^{\infty} \frac{P_n(c)}{n+1}= \sum_{n=0}^{\infty} \int_{0}^{1} P_n(c) t^n dt= \int_{0}^{1} \frac{dt}{\sqrt{1-2ct+t^2}}=\int_{0}^{1} \frac{dt}{\sqrt{(t -c)^2+1-c^2}}$$ $$\implies S=ln[2((t-c)+\sqrt{(t-c)^2+s^2})|_{0}^{1}=\ln\frac{1-c+\sqrt{2(1-c)}}{(1-c)}=\ln \frac{1+\sin(\theta/2)}{\sin(\theta/2)}.$$