\frac{\left(\frac{a\sqrt{2}}{2}\cdot \sqrt{2-\cos^2\alpha}\right)^2-a^2}{\left(\frac{a\sqrt{2}}{2}\cdot \sqrt{2-\cos^2\alpha}\right)^2}=
=\frac{\left(\frac{a\sqrt{2}}{2}\right)^2\cdot \left(2-\cos^2\alpha\right)-a^2}{\left(\frac{a\sqrt{2}}{2}\right)^2\cdot \left(2-\cos^2\alpha\right)}=
=\frac{\frac{a^2}{2}\cdot \left(2-\cos^2\alpha\right)-a^2}{\frac{a^2}{2}\cdot \left(2-\cos^2\alpha\right)}=
=\frac{\frac{a^2}{2}\cdot\Big( \left(2-\cos^2\alpha\right)-2\Big)}{\frac{a^2}{2}\cdot \left(2-\cos^2\alpha\right)}=
=\frac{2-\cos^2\alpha-2}{2-\cos^2\alpha}=
=\frac{-\cos^2\alpha}{2-\cos^2\alpha}= \hspace{8ex} \frac{|\cdot(-1)}{|\cdot(-1)}
=\frac{\cos^2\alpha}{-(2-\cos^2\alpha)}=
=\frac{\cos^2\alpha}{\cos^2\alpha-2}