If $( x = \sqrt {a^{\large sin^{-1}t}}, \: y = \sqrt {a^{\large cos^{-1}t}} ),$ what is the value of $\frac {\large dy}{\large dx}$

Question

$\begin{array}{1 1} (A) -\large\frac{y}{x} \\(B) -\large\frac{x}{y} \\ (C) \large\frac{y}{x} \\ (D) \large\frac{x}{y} \end{array} $