2. inverse trigonometric function — Maths STD 12 Science — Question

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

$\therefore \tan ^{-1} \frac{x}{\sqrt{a^{2}-x^{2}}}$

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

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

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

$=\tan ^{-1}(\tan \theta)=\theta=\sin ^{-1} \frac{x}{a}$