A possible value of

$\tan \left(\frac{1}{4} \sin ^{-1} \frac{\sqrt{63}}{8}\right)$

Let $\sin ^{-1}\left(\frac{\sqrt{63}}{8}\right)=\theta \quad \sin \theta=\frac{\sqrt{63}}{8}$

$\cos \theta=\frac{1}{8}$

$2 \cos ^{2} \frac{\theta}{2}-1=\frac{1}{8}$

$\cos ^{2} \frac{\theta}{2}=\frac{9}{16}$

$\cos \frac{\theta}{2}=\frac{3}{4}$

$\frac{1-\tan ^{2} \frac{\theta}{4}}{1+\tan ^{2} \frac{\theta}{4}}=\frac{3}{4}$

$\tan \frac{\theta}{4}=\frac{1}{\sqrt{7}}$