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Simplify a short product or sum of ratios to one ratio (tan + cot = sec cosec, sec - cos = sin tan)

This type of question, asked 4 times: 15 Nov 2020 Tier II, 29 Jan 2022 Tier II, 3 Feb 2022 Tier II, 8 Aug 2022 Tier II. All 6 questions below, oldest first.

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  1. Question 1

    15 Nov 2020 Tier II

    79
    cos⁡A(sec⁡A−cos⁡A)(cot⁡A+tan⁡A)=?\cos A(\sec A - \cos A)(\cot A + \tan A) = ?

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  2. Question 2

    29 Jan 2022 Tier II

    68
    Let 0∘<θ<90∘0^\circ < \theta < 90^\circ. (1+cot⁡2θ)(1+tan⁡2θ)×(sin⁡θ−cosec θ)(cos⁡θ−sec⁡θ)(1 + \cot^2\theta)(1 + \tan^2\theta) \times (\sin\theta - \text{cosec}\,\theta)(\cos\theta - \sec\theta) is equal to:

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  3. Question 3

    3 Feb 2022 Tier II

    27
    The expression (tan⁡θ+cot⁡θ)(sec⁡θ+tan⁡θ)(1−sin⁡θ)(\tan\theta + \cot\theta)(\sec\theta + \tan\theta)(1 - \sin\theta), 0∘<θ<90∘0^\circ < \theta < 90^\circ, is equal to:

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  4. Question 4

    8 Aug 2022 Tier II

    36
    Which of the following is equal to [cos⁡θsin⁡θ+sin⁡θcos⁡θ]\left[\frac{\cos\theta}{\sin\theta} + \frac{\sin\theta}{\cos\theta}\right]?

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  5. Question 5

    8 Aug 2022 Tier II

    54
    Which of the following is equal to 1tan⁡θ+tan⁡θ\frac{1}{\tan\theta} + \tan\theta ?

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  6. Question 6

    8 Aug 2022 Tier II

    30
    Which of the following is equal to sec⁡A−cos⁡A\sec A - \cos A?

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