If $\left[{\cos }^{-1}x\right]+\left[{\cot }^{-1}x\right]=0$ , where [.] denotes the greatest integer function, then complete set of values of $x$  is

$0\le {\cos }^{-1}x\le \pi \Rightarrow \left[{\cos }^{-1}x\right]=0,1,2,or3$

$0\le {\cos }^{-1}x\le \pi \Rightarrow \left[{\cot }^{-1}x\right]=0,1,2,or3$

$\Rightarrow \left[{\cos }^{-1}x\right]+\left[{\cot }^{-1}x\right]=0$

$\Rightarrow \left[{\cos }^{-1}x\right]=0,\left[{\cot }^{-1}x\right]=0$

$\Rightarrow 0\le {\cos }^{-1}x<1,0<{\cot }^{-1}x<1$

$\Rightarrow \cos 1<x\le 1,\cot 1<x<\infty$

Since $\cot 1>\cos 1$ , set of values of $x$  is $\left(\cot 1,1\right]$ .