Write the function in the simplest form:
Question:

Write the function in the simplest form:

$\tan ^{-1} \frac{x}{\sqrt{a^{2}-x^{2}}},|x|<a$

Solution:

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

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

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

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

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

 

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