Solution

$${ \sin }^{ -1 }\left| \sin kx \right| =kx\quad kx\epsilon \left[ 0,2\pi \right] $$ ...(1)
$${ cos }^{ -1 }\left( cos x \right) =kx\quad kx\epsilon \left[ 0,\pi \right] $$ ...(2)
From (1) and (2)
$${ \sin }^{ -1 }\left| \sin kx \right| +{ \cos }^{ -1 }\left( \cos x \right) $$ is constant
when $$kx\epsilon \left[ 0,\pi \right] $$
For for interval of $$\cfrac { \pi }{ 4 } $$
$$x=\left[ 0,\cfrac { \pi }{ 4 } \right] \Rightarrow k=4$$