$sin \left (sin^{-1}\ \frac{2}{5} + cos^{-1}\ x \right) =1$
$\implies \left (sin^{-1}\ \frac{2}{5} + cos^{-1}\ x \right) =sin^{-1}\ 1$
$\implies sin^{-1}\ \frac{2}{5} + cos^{-1}\ x =\frac{\pi}{2}$
$\implies cos^{-1}\ x =\frac{\pi}{2}- sin^{-1}\ \frac{2}{5}$
$\implies cos^{-1}\ x =cos^{-1}\ \frac{2}{5}$ $( \because sin^{-1}\ \frac{2}{5} + cos^{-1}\ \frac{2}{5}=\frac{\pi}{2})$
$\implies x= \frac{2}{5}$
$\therefore$ Option $B.$ is correct