The correct answer is $\boxed{\text{A}. \sin a \cos A}$.
In a spherical triangle, the sine of an angle is equal to the product of the cosines of the other two angles and the sine of the opposite side. In this case, the angle $b$ is opposite the side $c$, and the other two angles are $a$ and $C$, which is a right angle. Therefore, the sine of $b$ is equal to $\sin b = \cos a \cos C \sin c$.
The other options are incorrect because they do not take into account the fact that the angle $b$ is opposite the side $c$. Option $\text{B}. \cos a \sin A$ is the sine of the angle $A$, which is not opposite the side $c$. Option $\text{C}. \tan a \cot A$ is the tangent of the angle $A$, which is not equal to the sine of the angle $b$. Option $\text{D}. \cot A \tan A$ is the cotangent of the angle $A$, which is also not equal to the sine of the angle $b$.