A possible value of tan 14Sin−1⁡638 is

Let 14sin−1⁡638=θ Then sin 4θ=638⇒Cos 4θ=18⇒2Cos2 2θ−1=18⇒cos⁡2θ=34⇒2cos2⁡θ−1=34⇒cos2⁡θ=78⇒cos⁡θ=722⇒tan14Sin−1⁡638=tan⁡θ=17

A possible value of tan 14Sin−1⁡638 is

Let 14sin−1⁡638=θ Then sin 4θ=638⇒Cos 4θ=18⇒2Cos2 2θ−1=18⇒cos⁡2θ=34⇒2cos2⁡θ−1=34⇒cos2⁡θ=78⇒cos⁡θ=722⇒tan14Sin−1⁡638=tan⁡θ=17

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

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$\sqrt{7} - 1$

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

$2 \sqrt{2} - 1$

$\frac{1}{2 \sqrt{2}}$

answer is D.

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Detailed Solution

Let $\frac{1}{4} \sin^{- 1} \frac{\sqrt{63}}{8} = θ$ Then sin $4 θ = \frac{\sqrt{63}}{8} \Rightarrow$ Cos $4 θ = \frac{1}{8} \Rightarrow$ 2Cos²2 $θ - 1 = \frac{1}{8}$ $\Rightarrow \cos 2 θ = \frac{3}{4} \Rightarrow 2 \cos^{2} θ - 1 = \frac{3}{4} \Rightarrow \cos^{2} θ = \frac{7}{8} \Rightarrow \cos θ = \frac{\sqrt{7}}{2 \sqrt{2}}$