Question

The set of all values of $k$ for which $\left(\tan ^{-1} x \right)^{3}+\left(\cot ^{-1} x \right)^{3}= k \pi^{3}, x \in R$, 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. $\left[\frac{1}{32}, \frac{9}{8}\right)$

Answer 1

Answer: (A)

$Let S =\left(\tan ^{-1} x +\cot ^{-1} x \right)-3 \tan ^{-1} x \cdot \cot ^{-1} x \left(\tan ^{-1} x +\cot ^{-1} x \right)$
$=\frac{\pi^{3}}{8}-\frac{3 \pi}{2} \tan ^{-1} x \left(\frac{\pi}{2}-\tan ^{-1} x \right)$
$=\frac{3 \pi}{2}\left(\tan ^{-1} x -\frac{\pi}{4}\right)^{2}+\frac{\pi^{3}}{32}$
$\Rightarrow \frac{\pi^{3}}{32} \leq S <\frac{7}{8} \pi^{3}$
$=\frac{\pi^{3}}{32} \leq K \pi^{3}<\frac{7}{8} \pi^{3}$
$ \frac{1}{32} \leq K <\frac{7}{8}$