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Range or maximum of powers of sin^-1 x and cos^-1 x (or tan^-1 and cot^-1, sec^-1 and cosec^-1)

This type of question, asked 3 times: 14 Jun 2022, 22 Jan 2025, 21 Jan 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    14 Jun 2022

    68
    The set of all values of kk for which (tan⁡−1x)3+(cot⁡−1x)3=kπ3,x∈R\left( \tan^{- 1}x \right)^{3}+\left( \cot^{- 1}x \right)^{3}= k\pi^{3},x \in R, is the interval :

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  2. Question 2

    22 Jan 2025

    55
    Using the principal values of the inverse trigonometric functions the sum of the maximum and the minimum values of 16((sec⁡−1x)2+(csc⁡−1x)2)16\left( \left( \sec^{- 1}x \right)^{2}+\left( \csc^{- 1}x \right)^{2} \right) is :

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  3. Question 3

    21 Jan 2026 S2

    73
    Let the maximum value of (sin⁡−1x)2+(cos⁡−1x)2\left( \sin^{- 1}x \right)^{2} + \left( \cos^{- 1}x \right)^{2} for x∈[−32,12]x \in \left\lbrack - \frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}} \right\rbrack be mnπ2\frac{m}{n}\pi^{2}, where gcd(m,n)=1gcd(m,n) = 1. Then m+nm + n is equal to ____\_\_\_\_ .