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 $\left[\sin ^{-1} \cos ^{-1} \sin ^{-1} \tan ^{-1} x\right]=1$
$\Rightarrow 1 \leq \sin ^{-1} \cos ^{-1} \sin ^{-1} \tan ^{-1} x \leq \frac{\pi}{2}$
$\Rightarrow \sin 1 \leq \cos ^{-1} \sin ^{-1} \tan ^{-1} x \leq 1$
$\Rightarrow \cos \sin 1 \geq \sin ^{-1} \tan ^{-1} x \geq \cos 1$
$\Rightarrow \sin \cos \sin 1 \geq \tan ^{-1} x \geq \sin \cos 1$
$\Rightarrow \tan \sin \cos \sin 1 \geq x \geq \tan \sin \cos 1$
Hence $x \in[\tan \sin \cos 1, \tan \sin \cos \sin 1]$