AUSTRIAN-POLISH 2004 Problem 3

Solve the following system of equations in $\mathbb{R}$ where all square roots are non-negative:

$\begin{matrix} a - \sqrt{1-b^2} + \sqrt{1-c^2} = d b - \sqrt{1-c^2} + \sqrt{1-d^2} = a c - \sqrt{1-d^2} + \sqrt{1-a^2} = b d - \sqrt{1-a^2} + \sqrt{1-b^2} = c \end{matrix}$