MCQ
Write the function in the simplest form: $\tan ^{-1}\left(\frac{1}{\sqrt{x^{2}-1}}\right),|x|>1$
- ✓$\frac{\pi}{2}-\sec ^{-1} x $
- B$\frac{\pi}{2}+\sec ^{-1} x $
- C$\frac{\pi}{2} + cosec ^{-1} x $
- D$\frac{\pi}{2}-cosec ^{-1} x $
Put $x=cosec \theta \Rightarrow \theta=cosec^{-1} x$
$\therefore \tan ^{-1} \frac{1}{\sqrt{x^{2}-1}}$
$=\tan ^{-1} \frac{1}{\sqrt{\cos e c^{2} \theta-1}}$
$=\tan ^{-1}\left(\frac{1}{\cot \theta}\right)$
$=\tan ^{-1}(\tan \theta)$
$=\theta$
$=cosec ^{-1} x$
$=\frac{\pi}{2}-\sec ^{-1} x $ $\left[A s, \cos e c^{-1} x+\sec ^{-1} x=\frac{\pi}{2}\right]$
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where $[x]$ is the greatest integer less than or equal to $x$, then the value of $\alpha$ is :