Total number of positive integral value $$'n'$$ so that the equations $${\cos ^{ - 1}}x + {\left( {{{\sin }^{ - 1}}y} \right)^2} = \frac{{n{\pi ^2}}}{4}\,$$     and $${\left( {{{\sin }^{ - 1}}y} \right)^2} - {\cos ^{ - 1}}x = \frac{{{\pi ^2}}}{{16}}\,$$     are consistent, is equal to

Solution :
$$\eqalign{ & {\text{We have, }}2{\left( {{{\sin }^{ - 1}}y} \right)^2} = \frac{{4n + 1}}{{16}}{\pi ^2} \cr & \Rightarrow 0 \leqslant \frac{{4n + 1}}{{32}}{\pi ^2} \leqslant \frac{{{\pi ^2}}}{4} \cr & {\text{Also, }}2\left( {{{\cos }^{ - 1}}x} \right) = \frac{{4n - 1}}{{16}}{\pi ^2} \cr & \Rightarrow - \frac{1}{4} \leqslant n \leqslant \frac{7}{4} \cr & {\text{Also, }}2\left( {{{\cos }^{ - 1}}x} \right) = \frac{{4n - 1}}{{16}}{\pi ^2} \cr & \Rightarrow 0 \leqslant \frac{{4n - 1}}{{32}}{\pi ^2} \leqslant \pi \cr & \Rightarrow \frac{1}{4} \leqslant n \leqslant \frac{8}{\pi } + \frac{1}{4} \cr & \Rightarrow n = 1 \cr} $$