Passage \( \quad: \) It is given that \( A=\left(\tan ^{-1} x\right)^{3}+\left(\cot ^{-1} x\righ... VIDEO
Passage \( \quad: \) It is given that \( A=\left(\tan ^{-1} x\right)^{3}+\left(\cot ^{-1} x\right)^{3} \), where \( x0 \& B=\left(\cos ^{-1} t\right)^{2}+\left(\sin ^{-1} t\right)^{2} \),
where \( t \in\left[0, \frac{}{\sqrt{2}}\right] \& \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2} \) for \( -1 \leq x \leq 1 \& \) \( \tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2} \forall x \in R \) The interval in which \( A \) lie is
(a) \( \left[\frac{\pi^{3}}{8}, \frac{\pi^{3}}{2}\right) \)
(b) \( \left[\frac{\pi^{3}}{32}, \frac{\pi^{3}}{8}\right) \)
(c) \( \left(\frac{\pi^{3}}{10}, \frac{\pi^{3}}{5}\right) \)
(d) None of these
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