$$
\begin{aligned}
\because & \sin 18^{\circ}=\frac{\sqrt{5}-1}{4} \\
& x^{2}=4^{2}-(\sqrt{5}-1)^{2} \\
\Rightarrow & x=\sqrt{10+2 \sqrt{5}} \\
\Rightarrow & \cos 18^{\circ}=\frac{\sqrt{10+2 \sqrt{5}}}{4} \\
\Rightarrow & 2 \cos ^{2} 9-1=\frac{\sqrt{10+2 \sqrt{5}}}{4} \\
& \cos ^{2} 9=\frac{\sqrt{10+2 \sqrt{5}}+4}{8} \\
\Rightarrow & \sin ^{2} 81=\frac{4+\sqrt{10+2 \sqrt{5}}}{8}
\end{aligned}
$$
After squaring all the options available, we come to a conclusion that option (a) is correct.