\begin{align*} &\quad\; \abs{x + i \sqrt{1 - x^2} \cos{\theta}}\\ =& \sqrt{x^2 + (1 - x^2) \cos^2{\theta}}\\ \le& \sqrt{x^2 + (1 - x^2)}\\ =& 1. \end{align*}

$\mathcal{C}$