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Values of sin^-1(sin x), cos^-1(cos x), tan^-1(tan x) when x is outside the principal range

This type of question, asked 4 times: 2021 Paper 24, 27 Jun 2022, 31 Jan 2023, 31 Jan 2024. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 24

    62
    cos⁡−1(cos⁡(−5))+sin⁡−1(sin⁡(6))−tan⁡−1(tan⁡(12))\cos^{- 1}(\cos( - 5)) +\sin^{- 1}(\sin(6)) -\tan^{- 1}(\tan(12)) is equal to: (The inverse trigonometric functions take the principal values)

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  2. Question 2

    27 Jun 2022

    79
    sin⁡−1(sin⁡2π3)+cos⁡−1(cos⁡7π6)+tan⁡−1(tan⁡3π4)\sin^{- 1}\left( \sin\frac{2\pi}{3} \right)+\cos^{- 1}\left( \cos\frac{7\pi}{6} \right)+\tan^{- 1}\left( \tan\frac{3\pi}{4} \right) is equal to:

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  3. Question 3

    31 Jan 2023

    79
    If sin⁡−1α17+cos⁡−145−tan⁡−17736=0,0<α<13\sin^{- 1}\frac{\alpha}{17}+\cos^{- 1}\frac{4}{5}-\tan^{- 1}\frac{77}{36}= 0,0 < \alpha < 13, then sin⁡−1(sin⁡α)+cos⁡−1(cos⁡α)\sin^{- 1}(\sin\alpha) +\cos^{- 1}(\cos\alpha) is equal to

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  4. Question 4

    31 Jan 2024

    166
    If a=sin⁡−1(sin⁡(5))a =\sin^{- 1}(\sin(5)) and b=cos⁡−1(cos⁡(5))b =\cos^{- 1}(\cos(5)), then a2+b2a^{2}+b^{2} is equal to

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