Question

Express the expression is simplest from ${\tan }^{-1}\left(\frac{x}{a+\sqrt{a^2-x^2}}\right)$

Answer 1

given

${\tan }^{-1}\left(\frac{x}{a+\sqrt{a^2-x^2}}\right)$

let $x=a\sin \theta \Rightarrow \theta ={\sin }^{-1}\left(\frac{x}{a}\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{a\sin \theta }{a+\sqrt{a^2-a^2{\sin }^2\theta }}\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{a\sin \theta }{a(1+\sqrt{1-{\sin }^2\theta })}\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}{1+\sqrt{{\cos }^2\theta }}\right)$ we know $\begin{array}{c}1+\cos 2\theta =2{\cos }^2\theta \\ \sin 2\theta =2\sin \theta \cos \theta \end{array}$

$\Rightarrow {\tan }^{-1}\left(\frac{2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}{1+\cos \theta }\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}{2{\cos }^2\frac{\theta }{2}}\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{\sin \frac{\theta }{2}}{\cos \frac{\theta }{2}}\right)$

$\Rightarrow {\tan }^{-1}\left(\tan \frac{\theta }{2}\right)$

$\Rightarrow \frac{\theta }{2}\Rightarrow \frac{1}{2}{\sin }^{-1}\left(\frac{x}{a}\right)$