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tan^-1 of a nested surd ratio written as a sum or difference of two tan^-1

This type of question, asked 2 times: 15 Apr 2026 S1, 15 May 2026 S1. All 2 questions below, oldest first.

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  1. Question 1

    15 Apr 2026 S1

    67
    If tan⁡−1[5−261+6]=π3−tan⁡−1(k)\tan^{-1}\left[\dfrac{\sqrt{5 - 2\sqrt{6}}}{1 + \sqrt{6}}\right] = \dfrac{\pi}{3} - \tan^{-1}(k), then sec⁡−1(k)=…\sec^{-1}(k) = \ldots

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  2. Question 2

    15 May 2026 S1

    67
    If tan⁡−1[5+261−6]=a+tan⁡−1(2)\tan^{-1}\left[\dfrac{\sqrt{5 + 2\sqrt{6}}}{1 - \sqrt{6}}\right] = a + \tan^{-1}(\sqrt{2}), then csc⁡(a)=…\csc(a) = \ldots

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