# Logarithmic derivative of Jacobi theta 2 equals negative tangent + a sum of sines

The following formula holds: $$\dfrac{\vartheta_2'(u,q)}{\vartheta_2(u,q)} = -\tan(u)+4\displaystyle\sum_{k=1}^{\infty} (-1)^k \dfrac{q^{2k}}{1-q^{2k}} \sin(2ku),$$ where $\vartheta_2$ denotes the Jacobi theta 2, $\tan$ denotes the tangent, and $\sin$ denotes sine.