Answer
$\frac{\sqrt {1-x^2}-xy}{\sqrt {1+y^2}}$
Work Step by Step
Step 1. Let $u=cos^{-1}x, v=tan^{-1}y$, we have $cos(u)=x, tan(v)=y$ and
$sin(u)=\sqrt {1-x^2}, sin(v)=\frac{y}{\sqrt {1+y^2}}, cos(v)=\frac{1}{\sqrt {1+y^2}}$
Step 2. Use the Subtraction Formula, we have $sin(cos^{-1}x-tan^{-1}y)
=sin(u-v)=sin(u)cos(v)-cos(u)sin(v)=\sqrt {1-x^2}\times\frac{1}{\sqrt {1+y^2}}-x\frac{y}{\sqrt {1+y^2}}
=\frac{\sqrt {1-x^2}-xy}{\sqrt {1+y^2}}$
