A possible value of ((1)/(4) ^ - 1 63 8 ) is:

Let $\dfrac{1}{4}\sin^{- 1}\dfrac{\sqrt{63}}{8} = \theta$

$\sin 4\theta = \dfrac{\sqrt{63}}{8}$

$\cos 4\theta = \dfrac{1}{8}$

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

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

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

$2\cos^{2}\theta - 1 = \dfrac{3}{4}$

$\cos^{2}\theta = \dfrac{7}{8}$

$cos\theta = \dfrac{\sqrt{7}}{2\sqrt{2}}$

$tan\theta = \dfrac{1}{\sqrt{7}}$