MathNet 2013 Question 32

In a triangle $ABC$ with $B = 90^\circ$, $D$ is a point on the segment $BC$ such that the inradii of triangles $ABD$ and $ADC$ are equal. If $\angle ADB = \varphi$ then prove that $\tan^2(\varphi/2) = \tan(C/2)$. $\square$