Suppose that for some angles x and y, the equations ${\sin }^2\mathrm{x}+{\cos }^2\mathrm{y}=\frac{3\mathrm{a}}{2}$ and ${\cos }^2\mathrm{x}+{\sin }^2\mathrm{y}=\frac{{\mathrm{a}}^2}{2}$ hold simultaneously, The possible value of a is,

$\begin{array}{r}{\sin }^2\mathrm{x}+{\cos }^2\mathrm{y}=\frac{3\mathrm{a}}{2}---\left(1\right)\\ \text{ and }{\cos }^2\mathrm{x}+{\sin }^2\mathrm{y}=\frac{{\mathrm{a}}^2}{2}----\left(2\right)\end{array}$

Adding (1) and (2), we get

$\begin{array}{l}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2=\frac{3\mathrm{a}}{2}+\frac{{\mathrm{a}}^2}{2}\\ \text{ or }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}^2+3\mathrm{a}-4=0\\ \text{ or }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}(\mathrm{a}+4)(\mathrm{a}-1)=0\\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{a}=-4\text{ (rejected), }\mathrm{a}=1\end{array}$