The set of all values of k for which $\left(\tan ^{-1} \mathrm{x}\right)^{3}+\left(\cot ^{-1} \mathrm{x}\right)^{3}=\mathrm{k} \pi^{3}, \mathrm{x} \in \mathrm{R}$, is the interval :

(A) $\left[\frac{1}{32}, \frac{7}{8}\right)$

(B) $\left(\frac{1}{24}, \frac{13}{16}\right)$

(C) $\left[\frac{1}{48}, \frac{13}{16}\right]$

(D) $\left[\frac{1}{32}, \frac{9}{8}\right)$

JEE Main · 2022 · 24 Jun 2022 · Shift 1 · Inverse Trigonometric Functions

The set of all values of k for which

(tan^{- 1} x)^{3} + (cot^{- 1} x)^{3} = k π^{3}, x \in R

, is the interval :

Answer: (A)

Let

S = (tan^{- 1} x)^{3} + (cot^{- 1} x)^{3}

. Using

tan^{- 1} x + cot^{- 1} x = \frac{π}{2}

and the identity

a^{3} + b^{3} = (a + b)^{3} - 3 ab (a + b)

, we get

S = \frac{π ^{3}}{8} - \frac{3 π}{2} tan^{- 1} x (\frac{π}{2} - tan^{- 1} x) = \frac{3 π}{2} (tan^{- 1} x - \frac{π}{4})^{2} + \frac{π ^{3}}{32}

. Since

tan^{- 1} x \in (- \frac{π}{2}, \frac{π}{2})

, the minimum of

S

\frac{π ^{3}}{32}

(at

x = 1

) and as

x \to \pm \infty

S \to \frac{3 π}{2} \cdot \frac{π ^{2}}{16} + \frac{π ^{3}}{32} = \frac{7 π ^{3}}{8}

. Thus

k \in [\frac{1}{32}, \frac{7}{8})

JEE Main · 2022 · 24 Jun 2022 · Shift 1 · Single correct option