MCQ
Write the function in the simplest form: $\tan ^{-1} \frac{x}{\sqrt{a^{2}-x^{2}}},|x| < a$
- A$\tan ^{-1} \frac{a}{x}$
- B$\tan ^{-1} \frac{x}{a}$
- C$\sin ^{-1} \frac{a}{x}$
- ✓$\sin ^{-1} \frac{x}{a}$
✓
Answer
Correct option: D.
$\sin ^{-1} \frac{x}{a}$
d
$\tan ^{-1} \frac{x}{\sqrt{a^{2}-x^{2}}}$
$\tan ^{-1} \frac{x}{\sqrt{a^{2}-x^{2}}}$
Let, $x=a \sin \theta \Rightarrow \frac{x}{a}=\sin \theta \Rightarrow \sin ^{-1}\left(\frac{x}{a}\right)$
$\therefore \tan ^{-1} \frac{x}{\sqrt{a^{2}-x^{2}}}$
$=\tan ^{-1}\left(\frac{a \sin \theta}{\sqrt{a^{2}-a^{2} \sin ^{2} \theta}}\right)$
$=\tan ^{-1}\left(\frac{a \sin \theta}{a \sqrt{1-\sin ^{2} \theta}}\right)$
$=\tan ^{-1}\left(\frac{a \sin \theta}{a \cos \theta}\right)$
$=\tan ^{-1}(\tan \theta)=\theta=\sin ^{-1} \frac{x}{a}$
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