Question

If $\left[{\sin }^{-1}{\cos }^{-1}{\sin }^{-1}{\tan }^{-1}x\right]=1$, where $[.]$ denotes the greatest integer function, then $x$ belongs to the interval

• A. $[\tan \sin \cos 1,\tan \sin \cos \sin 1]$
• B. $[\tan \sin \cos 1,\tan \sin \cos \sin 2]$
• C. $[-1,1]$
• D. $[\sin \cos \tan 1,\sin \cos \sin \tan 1]$

Answer 1

Answer: (A)

We have $\left[{\sin }^{-1}{\cos }^{-1}{\sin }^{-1}{\tan }^{-1}x\right]=1$
$\Rightarrow 1\le {\sin }^{-1}{\cos }^{-1}{\sin }^{-1}{\tan }^{-1}x\le \frac{\pi }{2}$
$\Rightarrow \sin 1\le {\cos }^{-1}{\sin }^{-1}{\tan }^{-1}x\le 1$
$\Rightarrow \cos \sin 1\ge {\sin }^{-1}{\tan }^{-1}x\ge \cos 1$
$\Rightarrow \sin \cos \sin 1\ge {\tan }^{-1}x\ge \sin \cos 1$
$\Rightarrow \tan \sin \cos \sin 1\ge x\ge \tan \sin \cos 1$
Hence $x\in [\tan \sin \cos 1,\tan \sin \cos \sin 1]$