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Inverse trig of a trig expression that simplifies to a linear function of x

This type of question, asked 5 times: May 2021 S1, 26 Apr 2025 S1, 15 Apr 2026 S1, 15 Apr 2026 S2, 19 Apr 2026 S1. All 5 questions below, oldest first.

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  1. Question 1

    May 2021 S1

    140
    If y=tan⁡−1 ⁣1+cos⁡x1−cos⁡xy = \tan^{-1}\!\sqrt{\frac{1+\cos x}{1-\cos x}}, then dydx=\frac{dy}{dx} =

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  2. Question 2

    26 Apr 2025 S1

    149
    If y=tan⁡−1(1+sin⁡x1−sin⁡x),0⩽x<π2y=\tan^{- 1}\left( \sqrt{\frac{1 + \sin x}{1 - \sin x}} \right),0 \leqslant x<\frac{\pi}{2}, then y′(π6)=y^{'}\left( \frac{\pi}{6} \right)=

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  3. Question 3

    15 Apr 2026 S1

    71
    If y=cos⁡−1(sin⁡x)y = \cos^{-1}(\sin x), where π2<x<π\dfrac{\pi}{2} < x < \pi, then dydx=…\dfrac{dy}{dx} = \ldots

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  4. Question 4

    15 Apr 2026 S2

    69
    If y=tan⁡−11+sin⁡2x1−sin⁡2xy = \tan^{-1}\sqrt{\dfrac{1 + \sin 2x}{1 - \sin 2x}}, then dydx\dfrac{dy}{dx} at x=π6x = \dfrac{\pi}{6} is

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  5. Question 5

    19 Apr 2026 S1

    73
    If y=cot⁡−1(1+sin⁡5xcos⁡5x)y = \cot^{-1}\left(\dfrac{1 + \sin 5x}{\cos 5x}\right), then the value of dydx\dfrac{dy}{dx} is

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