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Maximum and minimum of tan^-1(sin x + cos x) or tan^-1(sin x - cos x) on an interval

The same question, asked 2 times: 2021 Paper 21*, 28 Jul 2022. Every wording below, oldest first.

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  1. Question 1

    2021 Paper 21

    63
    Let MM and mm respectively be the maximum and minimum values of the function f(x)=tan⁡−1(sin⁡x+cos⁡x)f(x) =\tan^{- 1}(\sin x+ \cos x) in [0,π2]\left\lbrack 0,\frac{\pi}{2} \right\rbrack, Then the value of tan⁡(M−m)\tan(M-m) is equal to:

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  2. Question 2

    28 Jul 2022

    157
    The sum of the absolute maximum and absolute minimum values of the function f(x)=tan⁡−1(sin⁡x−cos⁡x)f(x) =\tan^{- 1}(\sin x - \cos x) in the interval [0,π]\lbrack 0,\pi\rbrack is

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