If the distance between the points $\left(\mathrm{acos}{48}^0,0\right)$ and $\left(0, \mathrm{acos}{12}^0\right)$ is $\mathrm{d},$ then ${\mathrm{d}}^2-{\mathrm{a}}^2=$

$$\begin{gathered} \mathrm{d}=\sqrt{{(\mathrm{a} \cos {48}^{\circ })}^2+{(-\mathrm{a} \cos {12}^{\circ })}^2} \\ d^2={\mathrm{a}}^2 {\cos }^2{48}^{\circ }+{\mathrm{a}}^2 {\cos }^2{12}^{\circ } \\ \mathrm{Now} {\mathrm{d}}^2-{\mathrm{a}}^2={\mathrm{a}}^2[{\cos }^{2}48+{\cos }^{2}12-1] \\ ={\mathrm{a}}^2[{\cos }^2{48}^0-{\sin }^{2}12] \\ ={\mathrm{a}}^2 \cos \left({60}^0\right) \cos {36}^0 (\because {\cos }^2\mathrm{A}-{\sin }^{{ }^2}\mathrm{B}=\cos (\mathrm{A}+\mathrm{B}\left)\cos \right(\mathrm{a}-\mathrm{B}\left)\right) \\ ={\mathrm{a}}^2\times \frac{1}{2}\times \frac{\sqrt{5}+1}{4} \\ ={\mathrm{a}}^2\left(\frac{\sqrt{5}+1}{8}\right) \end{gathered}$$