$If the probability that$${\sin }^{-1}\left(\sin x\right)+{\cos }^{-1}\left(\cos y\right)$$is an integer$ $where$ $x,y\in \{1,2,3,4\}$  $is k, then 4k =$

$For expression$${\sin }^{-1}\left(\sin x\right)+{\cos }^{-1}\left(\cos y\right)$ $to be an integer,$ $x$ $should lie between$

$$\begin{gathered} \left[-\frac{\pi }{2},\frac{\pi }{2}\right]\text{ and }y\text{ should lie between }[0,\pi ]. \\ \Rightarrow x=1 \text{and }y=1,2,3 \end{gathered}$$

$\text{∴Required probability }=\frac{3}{16}=k\Rightarrow 4k=\frac{3}{4}=0.75$