The value(s) of θ satisfying the equation [ ( ^ -1 ^ -1 x)]= θ ,...

We have $\sin \left(\cot ^{-1} \cos \tan ^{-1} x\right)=\sin \left(\cot ^{-1} \cos \theta\right)$, where $\theta=\tan ^{-1} x=\sin \cot ^{-1}$\frac{1}{\sqrt{1+\tan ^2 \theta}}=\sin \cot ^{-1} \frac{1}{\sqrt{1+x^2}}=\sqrt{\frac{1+x^2}{2+x^2}} Now for all $x \in \mathbb{R}, 0 \lt$\frac{1+x^2}{2+x^2} \lt$1$, then \left[\sin \left(\cot ^{-1} \cos \tan ^{-1} x\right)\right]=\left[\frac{1+x^2}{2+x^2}\right]=$0\therefore$ The equation is $\sin \theta=0 \Rightarrow \theta=\mathrm{n} \pi, \mathrm{n} \in \mathrm{I}$