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sin(cot^-1(x+1)) = cos(tan^-1 x): find x

The same question, asked 2 times: 4 May 2023 S1*, 12 May 2024 S2. Every wording below, oldest first.

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  1. Question 1

    4 May 2023 S1

    114
    If sin⁡(cot⁡−1(x+1))=cos⁡(tan⁡−1x)\sin(\cot^{-1}(x+1))=\cos(\tan^{-1}x), then considering positive square roots, xx has the value

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  2. Question 2

    12 May 2024 S2

    113
    The value of xx, for which sin⁡ ⁣(cot⁡−1(x))=cos⁡ ⁣(tan⁡−1(1+x))\sin\!\left(\cot^{-1}(x)\right) = \cos\!\left(\tan^{-1}(1+x)\right), is

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