Question

A possible value of $\tan \left(\frac{1}{4}{\sin }^{-1}\frac{\sqrt{63}}{8}\right)$ is :

• A. $\frac{1}{\sqrt{7}}$
• B. $2\sqrt{2}-1$
• C. $\sqrt{7}-1$
• D. $\frac{1}{2\sqrt{2}}$

Answer 1

Answer: (A)

Let $\frac{1}{4}{\sin }^{-1}\frac{\sqrt{63}}{8}=\theta$
$\sin 4\theta =\frac{\sqrt{63}}{8}$
$\cos 4\theta =\frac{1}{8}$
$2{\cos }^{2}2\theta -1=\frac{1}{8}$
${\cos }^{2}2\theta =\frac{9}{16}$
$\cos 2\theta =\frac{3}{4}$
$2{\cos }^2\theta -1=\frac{3}{4}$
${\cos }^2\theta =\frac{7}{8}$
$\cos \theta =\frac{\sqrt{7}}{2\sqrt{2}}$
$\tan \theta =\frac{1}{\sqrt{7}}$