Logarithm of quotient of Jacobi theta 1 equals the log of a quotient of sines + a sum of sines

The following formula holds: $$\log \left( \dfrac{\vartheta_1(\alpha+\beta,q)}{\vartheta_1(\alpha - \beta,q)} \right)=\log \left( \dfrac{\sin(\alpha+\beta)}{\sin(\alpha-\beta)} \right) +4 \displaystyle\sum_{k=1}^{\infty} \dfrac{q^{2k}}{1-q^{2k}}\sin(2k\alpha)\sin(2k\beta),$$ where $\log$ denotes the logarithm, $\vartheta_1$ denotes the Jacobi theta 1, and