Power Series Expansion for Logarithm of Sine of x

Theorem
for all $x \in \R$ such that $0 < \left|{x}\right| < \pi$.

Proof
From Power Series Expansion for Cotangent Function:

for $0 < \left|{x}\right| < \pi$.

From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term: