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Trig x + 1/x = k (sec + cos, tan + cot): find x^2 + 1/x^2

This type of question, asked 3 times: 14 Jul 2023 S4, 26 Jul 2023 S3, 24 Sep 2024 S2. All 3 questions below, oldest first.

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  1. Question 1

    14 Jul 2023 S4

    67
    If sec⁡θ+cos⁡θ=32\sec\theta + \cos\theta = 32, then sec⁡2θ+cos⁡2θ\sec^2\theta + \cos^2\theta is _________.

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  2. Question 2

    26 Jul 2023 S3

    54
    If cos⁡x+sec⁡x=723\cos x + \sec x = \frac{7}{2\sqrt{3}}, then the value of cos⁡2x+sec⁡2x\cos^2 x + \sec^2 x will be ______.

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  3. Question 3

    24 Sep 2024 S2

    69
    If tan⁡A+cot⁡A=2\tan A + \cot A = 2, then the value of tan⁡2A+cot⁡2A\tan^2 A + \cot^2 A is:

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