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Fractions over 1 + sin and 1 - sin (or cos): add, then simplify or solve

This type of question, asked 3 times: 19 Jul 2023 S1, 12 Sep 2024 S2, 13 Sep 2024 S1. All 3 questions below, oldest first.

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  1. Question 1

    19 Jul 2023 S1

    74
    If (cos⁡A1−sin⁡A)+(cos⁡A1+sin⁡A)=4\left(\frac{\cos A}{1 - \sin A}\right) + \left(\frac{\cos A}{1 + \sin A}\right) = 4, then what will be the value of A? (0∘<θ<90∘)(0^\circ < \theta < 90^\circ)

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  2. Question 2

    12 Sep 2024 S2

    58
    Solve the following: 11+cos⁡θ+11−cos⁡θ=?\frac{1}{1 + \cos\theta} + \frac{1}{1 - \cos\theta} = ?

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  3. Question 3

    13 Sep 2024 S1

    59
    If 11−sin⁡θ+11+sin⁡θ=4sec⁡θ, (0<θ<90∘)\frac{1}{1 - \sin\theta} + \frac{1}{1 + \sin\theta} = 4\sec\theta,\ (0 < \theta < 90^\circ), then the value of (cot⁡θ+cosec θ)(\cot\theta + \text{cosec}\,\theta) is:

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