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)

$\frac{1}{\sqrt{7}}$

Answer 2

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