Write each of the following in the simplest form:

`tan^(-1){sqrt((a-x)/(a+x))}`

Explanation

Let `x=a \cos \theta`
Now
$$
\begin{aligned}
&\tan ^{-1} \sqrt{\frac{a-x}{a+x}}=\tan ^{-1} \sqrt{\frac{a-a \cos \theta}{a+a \cos \theta}} \\
&=\tan ^{-1} \sqrt{\frac{1-\cos \theta}{1+\cos \theta}} \\
&=\tan ^{-1} \sqrt{\frac{2 \sin ^{2} \frac{\theta}{2}}{2 \cos ^{2} \frac{\theta}{2}}} \\
&=\tan ^{-1}\left(\tan \frac{\theta}{2}\right) \\
...