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.
$$\left[ {\tan \sin \cos 1,\tan \sin \cos \sin 1} \right]$$
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 :
$$\left[ {\tan \sin \cos 1,\tan \sin \cos \sin 1} \right]$$
Solution :
$$\eqalign{
& {\text{We have}},1 \leqslant {\sin ^{ - 1}}{\cos ^{ - 1}}{\sin ^{ - 1}}{\tan ^{ - 1}}x \leqslant \frac{\pi }{2} \cr
& \Rightarrow \sin 1 \leqslant {\cos ^{ - 1}}{\sin ^{ - 1}}{\tan ^{ - 1}}x \leqslant 1 \cr
& \Rightarrow \cos \sin 1 \geqslant {\sin ^{ - 1}}{\tan ^{ - 1}}x \geqslant \cos 1 \cr
& \Rightarrow \sin \cos \sin 1 \geqslant {\tan ^{ - 1}}x \geqslant \sin \cos 1 \cr
& \Rightarrow \tan \sin \cos \sin 1 \geqslant x \geqslant \tan \sin \cos 1 \cr
& \therefore x \in \left[ {\tan \sin \cos 1,\tan \sin \cos \sin 1} \right] \cr} $$