JEE Main 2022 (Online) 24th June Morning ShiftQuestion 1
The set of all values of $k$ for which $(\tan^{-1}x)^{3}+(\cot^{-1}x)^{3}=k\pi^{3}$, $x\in\mathbf{R}$, is the interval :
Inverse Trigonometric Functions
Solution
Let $t=\tan^{-1}x\in(-\frac\pi2,\frac\pi2)$, $\cot^{-1}x=\frac\pi2-t$. $t^3+(\frac\pi2-t)^3=\frac{\pi^3}{8}-\frac{3\pi}{2}\,t(\frac\pi2-t)$. $t(\frac\pi2-t)$ has max $\frac{\pi^2}{16}$ at $t=\frac\pi4$ and tends to $-\frac{\pi^2}{2}$ as $t\to-\frac\pi2$ (not attained). So value ranges over $[\frac{\pi^3}{32},\frac{7\pi^3}{8})$, i.e. $k\in[\frac1{32},\frac78)$. Option (A).