A possible value of tan 1 4 sin - 1 63 8 is:

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Question: Answered & Verified by Expert

A possible value of

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

is:

MathematicsInverse Trigonometric FunctionsJEE MainJEE Main 2021 (24 Feb Shift 2)

Options:

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

Solution:

2338 Upvotes Verified Answer

The correct answer is:

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

Let $\frac{1}{4} \sin^{- 1} \frac{\sqrt{63}}{8} = θ$

$\sin 4 θ = \frac{\sqrt{63}}{8}$

$\cos 4 θ = \frac{1}{8}$

$2 \cos^{2} 2 θ - 1 = \frac{1}{8}$

$\cos^{2} 2 θ = \frac{9}{16}$

$\cos 2 θ = \frac{3}{4}$

$2 \cos^{2} θ - 1 = \frac{3}{4}$

$\cos^{2} θ = \frac{7}{8}$

$\cos θ = \frac{\sqrt{7}}{2 \sqrt{2}}$

$\tan θ = \frac{1}{\sqrt{7}}$

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