If $\left[{\sin }^{-1}{\cos }^{-1}{\sin }^{-1}{\tan }^{-1}\mathrm{\theta }\right]=1$, where [.] denotes the greatest integer function, the θ lies in the interval

We have, $\left[{\sin }^{-1}{\cos }^{-1}{\sin }^{-1}{\tan }^{-1}\mathrm{\theta }\right]=1$

$\begin{array}{l}\Rightarrow 1\le {\sin }^{-1}{\cos }^{-1}{\sin }^{-1}{\tan }^{-1}\mathrm{\theta }\le \frac{\mathrm{\pi }}{2}\\ \Rightarrow \sin 1\le {\cos }^{-1}{\sin }^{-1}{\tan }^{-1}\mathrm{\theta }\le 1\\ \Rightarrow \cos \sin 1\ge {\sin }^{-1}{\tan }^{-1}\mathrm{\theta }\ge \cos 1\\ \Rightarrow \sin \cos \sin 1\ge {\tan }^{-1}\mathrm{\theta }\ge \sin \cos 1\\ \Rightarrow \tan \sin \cos \sin 1\ge \mathrm{\theta }\ge \tan \sin \cos 1\end{array}$