\[
\begin{aligned}
&\sin ^{2} x+\cos ^{2} x=1 \\
&\sin ^{4} x+\cos ^{4} x+2 \sin ^{2} x \cos ^{2} x=1 \\
&\sin ^{4} x+\cos ^{4} x=1-2 \sin ^{2} x \cos ^{2} x \quad \ldots[1] \\
&\sin ^{2} x+\cos ^{2} x=1 \\
&\sin ^{6} x+\cos ^{6} x+3 \sin ^{2} x \cos ^{2} x\left(\sin ^{2} x+\cos ^{2} x\right)=1 \\
&\sin ^{6} x+\cos ^{6} x=1-3 \sin ^{2} x \cos ^{2} x \quad \ldots[2] \\
&\mid \sqrt{2 \sin ^{4} x+18 \cos ^{2} x}-\sqrt{2 \cos ^{4} x+18 \sin ^{2} x}=1
\end{aligned}
\]
Squaring above equation
\[
\begin{aligned}
&\therefore 2 \sin ^{4} x+18 \cos ^{2} x+2 \cos ^{4} x+18 \sin ^{2} x- \\
&2\left(\sqrt{2 \sin ^{4} x+18 \cos ^{2} x}\right)\left(\sqrt{2 \cos ^{4} x+18 \sin ^{2} x}\right)=1 \\
&\therefore 2\left(\sin ^{4} x+\cos ^{4} x\right)+18\left(\cos ^{2} x+\sin ^{2} x\right)-1= \\
&2\left(\sqrt{2 \sin ^{4} x+18 \cos ^{2} x}\right)\left(\sqrt{2 \cos ^{4} x+18 \sin ^{2} x}\right)
\end{aligned}
\]
\(\therefore 2\left(1-2 \sin ^{2} x \cos ^{2} x\right)+18-1=2\left(\sqrt{2 \sin ^{4} x+18 \cos ^{2} x}\right)\left(\sqrt{2 \cos ^{4} x+18 \sin ^{2} x}\right)\)
\(. .(\) [Using[1] )
\(\therefore 19-4 \sin ^{2} x \cos ^{2} x=2\left(\sqrt{2 \sin ^{4} x+18 \cos ^{2} x}\right)\left(\sqrt{2 \cos ^{4} x+18 \sin ^{2} x}\right)\)
Squaring above equation
\(\therefore 361+16 \sin ^{4} x \cos ^{4} x-152 \sin ^{2} x \cos ^{2} x=4\left(4 \sin ^{4} x \cos ^{4} x+36 \cos ^{6} x+36 \sin ^{6} x+\right.\)
\(324 \sin ^{2} x \cos ^{2} x\) )
\(\therefore 361+16 \sin ^{4} x \cos ^{4} x-152 \sin ^{2} x \cos ^{2} x=16 \sin ^{4} x \cos ^{4} x+144 \cos ^{6} x+144 \sin ^{6} x+\) \(1296 \sin ^{2} x \cos ^{2} x\)
\(\therefore 361=144\left(\cos ^{6} x+\sin ^{6} x\right)+1448 \sin ^{2} x \cos ^{2} x\)
\(\therefore 361=144\left(1-3 \sin ^{2} x \cos ^{2} x\right)+1448 \sin ^{2} x \cos ^{2} x\)
(Using [2] )
\(\therefore 217=1016 \sin ^{2} x \cos ^{2} x\)
\begin{aligned}
&\therefore 217=1016 \sin ^{2} \mathrm{x} \cos ^{2} \mathrm{x} \\
&\therefore 217=254 \sin ^{2} 2 \mathrm{x} \\
&\therefore \sin 2 \mathrm{x}=\pm \sqrt{\frac{217}{254}} \\
&\text { Let } \sin ^{-1} \sqrt{\frac{217}{254}}=\theta \\
&\sin 2 \mathrm{x}=\pm \sqrt{\frac{217}{254}} \\
&\therefore 2 \mathrm{x}=\sin ^{-1} \sqrt{\frac{217}{254}} \\
&\therefore 2 \mathrm{x}=\theta, \pi-\theta, 2 \pi+\theta, 3 \pi-\text { theta } \\
&\therefore \mathrm{x}=\frac{\pi}{2}, \frac{\pi-\theta}{2}, \frac{2 \pi+\theta}{2}, \frac{3 \pi-\theta}{2}
\end{aligned}
\[
2 x=\sin ^{-1}\left(-\sqrt{\frac{217}{254}}\right)
\]
\[
\therefore 2 \mathrm{x}=\pi+\theta, 2 \pi-\theta, 3 \pi+\theta, 4 \pi-\theta
\]
\[
\therefore \mathrm{x}=\frac{\pi+\theta}{2}, \frac{2 \pi-\theta}{2}, \frac{3 \pi+\theta}{2}, \frac{4 \pi-\theta}{2}
\]
So, total number of solutions \(=8\)