Answer
$$\frac{\sqrt{a^2+x^2}}{\pm x}$$
Work Step by Step
We have,$$\csc \left(\arctan \frac{x}{a}\right)$$
Now, solving by using the methods of inverse trigonometric functions:
\begin{aligned}
& \arctan \left(\frac{x}{a}\right)=\theta \\
& \Rightarrow \tan \theta=\frac{x}{a} \\
\Rightarrow & \cot \theta=\frac{a}{x}, x \neq 0 \\
\Rightarrow & \csc ^{2} \theta=1+\cot ^{2} \theta \\
=& 1+\frac{a^{2}}{x^{2}} \\
=& \frac{x^{2}+a^{2}}{x^{2}} \\
\Rightarrow & \csc \theta=\frac{\sqrt{a^{2}+x^{2}}}{\pm x} \\
& \csc \left(\arctan \left(\frac{x}{a}\right)\right) \\
=& \frac{\sqrt{a^{2}+x^{2}}}{\pm x}, x \neq 0
\end{aligned}
