If the distance between the points $A\left(a\sin {42}^0,0\right)$ and $B\left(0,a\sin {78}^0\right)$ is $d$ then $d^2-a^2=$

$\begin{array}{l}A=\left(a\sin {42}^{\circ },0\right)B=\left(0,a\sin {78}^{\circ }\right)\\ d=AB=\sqrt{a^2{\sin }^2\left(42\right)+a^2{\sin }^2\left(78\right)}\\ \frac{d^2}{a^2}={\sin }^2{42}^{\circ }+{\sin }^2{78}^{\circ }\\ ={\cos }^2{48}^{\circ }+{\cos }^2{12}^{\circ }={\cos }^2{48}^{\circ }+\left(1-{\sin }^2{12}^{\circ }\right)\end{array}$
$\begin{array}{l}\frac{d^2}{a^2}-1=\cos {60}^{\circ }\cos {36}^{\circ }\\ \frac{d^2-a^2}{a^2}=\frac{1}{2}\left(\frac{\sqrt{5}+1}{4}\right)\\ d^2-a^2=a^2\left(\frac{\sqrt{5}+1}{8}\right)\end{array}$