Consider a right-angled triangle \(\triangle A B C\) whose hypotenuse \(A C\) is of length 1. The bisector of \(\angle A C B\) intersects \(A B\) at \(D\). If \(B C\) is of length \(x\), then what is the length of \(C D\) ?
(A) \(\sqrt{\frac{2 x^2}{1+x}}\)
(B) \(\frac{1}{\sqrt{2+2 x}}\)
(c) \(\sqrt{\frac{x}{1+x}}\)
(D) \(\frac{x}{\sqrt{1-x^2}}\)