Let $x+\frac{1}{x}=\mathrm{2,}$ $y+\frac{1}{y}=-2$ and ${\sin }^{-1}x+{\text{cos}}^{-1}y=m\mathrm{\pi ,}$ then the value of $m$ is

$x+\frac{1}{x}=2\Rightarrow x=1$ $\dots .[x+\frac{1}{x}\ge 2$ as equality occurs only when $x=1]$
$\Rightarrow {\mathrm{s}\mathrm{i}\mathrm{n}}^{-1}x=\frac{\pi }{2}$ 

Similarly, $y=-1$ $....[y+\frac{1}{y}\le -2$ as equality occurs only when $y=-1]$
$\Rightarrow {\mathrm{c}\mathrm{o}\mathrm{s}}^{-1}y=\pi$
${\mathrm{s}\mathrm{i}\mathrm{n}}^{-1}x+{\mathrm{c}\mathrm{o}\mathrm{s}}^{-1}y=\frac{\pi }{2}+\pi$
$=\frac{3\pi }{2}$
$\Rightarrow m=\frac{3}{2}=1.5$