Answer
$csc~(arctan~\frac{\sqrt{9-u^2}}{u}) = \frac{3~\sqrt{9-u^2}}{9-u^2}$
Work Step by Step
Let $~~\theta = arctan~\frac{\sqrt{9-u^2}}{u}$
Then $~~tan~\theta = \frac{\sqrt{9-u^2}}{u}$
We can find an expression for $csc~\theta$:
$csc~\theta = \frac{\sqrt{(\sqrt{9-u^2})^2+u^2}}{\sqrt{9-u^2}}$
$csc~\theta = \frac{\sqrt{9-u^2+u^2}}{\sqrt{9-u^2}}$
$csc~\theta = \frac{\sqrt{9}}{\sqrt{9-u^2}}$
$csc~\theta = \frac{3}{\sqrt{9-u^2}}$
$csc~\theta = (\frac{3}{\sqrt{9-u^2}})~(\frac{\sqrt{9-u^2}}{\sqrt{9-u^2}})$
$csc~\theta = \frac{3~\sqrt{9-u^2}}{9-u^2}$
Therefore, $~~csc~(arctan~\frac{\sqrt{9-u^2}}{u}) = \frac{3~\sqrt{9-u^2}}{9-u^2}$