Question

If $\left[{\sin }^{-1}\left({\cos }^{-1}\left({\sin }^{-1}\left({\tan }^{-1}x\right)\right)\right)\right]=1$, where $[$ . $]$ denotes the greatest integer function then the values of $x$ are

• A. $\left[\tan \sin \cos 1,\tan \sin \cos \sin 1\right]$
• B. $\left(\tan \sin \cos 1,\tan \sin \cos \sin 1\right)$
• C. $\left[-1,1\right]$
• D. $\left[\sin \cos \tan 1,\sin \cos \sin \tan 1\right]$

Answer 1

The correct option is A $\left[\tan \sin \cos 1,\tan \sin \cos \sin 1\right]$

$\left[{sin}^{-1}\left({cos}^{-1}\left({sin}^{-1}\left({tan}^{-1}x\right)\right)\right)\right]=1$

then

$1\le {sin}^{-1}\left({cos}^{-1}\left({sin}^{-1}\left({tan}^{-1}x\right)\right)\right)\le \frac{\Pi }{2}sin1\le {cos}^{-1}\left({sin}^{-1}\left({tan}^{-1}x\right)\right)\le 1cossin1\le {sin}^{-1}\left({tan}^{-1}x\right)\le cos1$

$sincossin1\le {tan}^{-1}x\le sincos1tansincossin1\le x\le tansincos1buttansincossin\ge tansincos1sox\in [tansincos1,tansincossin1]Aiscorrect$