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

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Inverse Trigonometric Functions

A possible value of

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

is:

Answer: (A)

Let

θ = sin^{- 1} \frac{63}{8}

, so

sin θ = \frac{63}{8} = \frac{3 7}{8}

.

Then

cos θ = 1 - sin^{2} θ = 1 - \frac{63}{64} = \frac{1}{8}

.

Using the half-angle formula:

tan \frac{θ}{2} = \frac{s i n θ}{1 + c o s θ} = \frac{3 7 /8}{1 + 1/8} = \frac{3 7 /8}{9/8} = \frac{7}{3}

For

tan \frac{θ}{4}

, apply the half-angle formula again:

tan \frac{θ}{4} = \frac{1 - c o s ( θ /2 )}{s i n ( θ /2 )}

From

tan \frac{θ}{2} = \frac{7}{3}

, we get

sin \frac{θ}{2} = \frac{7}{4}

and

cos \frac{θ}{2} = \frac{3}{4}

.

Thus:

tan \frac{θ}{4} = \frac{1 - 3/4}{7 /4} = \frac{1/4}{7 /4} = \frac{1}{7}

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Single correct option