JEE Mains · Maths · STD 12 - 2. inverse trigonometric function
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 & Solution
Correct Answer
(A) \(\left[\frac{1}{32}, \frac{7}{8}\right)\)
Step-by-step Solution
Detailed explanation
Let \(S =\left(\tan ^{-1} x \right)^{3}+\left(\cot ^{-1} x \right)^{3}\) \(\left(\tan ^{-1} x+\cot ^{-1} x\right)-3 \tan ^{-1} x \cdot \cot ^{-1} x\left(\tan ^{-1} x+\cot ^{-1} x\right)\) \(=\frac{\pi^{3}}{8}-\frac{3 \pi}{2} \tan ^{-1} x\left(\frac{\pi}{2}-\tan ^{-1} x\right)\)…
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