The range of the value of p for which the equation $\sin \left({\cos }^{-1}\left(\cos \left({\tan }^{-1}x\right)\right)\right)=p$has a solution is

${\mathrm{sincos}}^{-1}\left(\cos \left({\tan }^{-1}x\right)\right)=p$

For $x\in R,{\tan }^{-1}x\in (-\pi /2,\pi /2)$

$\Rightarrow \cos \left({\tan }^{-1}x\right)\in (0,1]$

$\Rightarrow {\cos }^{-1}\cos \left({\tan }^{-1}x\right)\in [0,\pi /2)$

$$\begin{gathered} \Rightarrow \sin \left[{\cos }^{-1}\left(\cos \left({\tan }^{-1}x\right)\right)\right]\in [0,1) \\ \Rightarrow p\in [0,1) \end{gathered}$$