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Ratios like (tan t + sec t - 1)/(tan t - sec t + 1) simplify to sec t + tan t (or cosec t + cot t)

This type of question, asked 6 times: CDS 2 2016, CDS 1 2020, CDS 1 2022, CDS 2 2022, CDS 1 2024, CDS 2 2024. All 6 questions below, oldest first.

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  1. Question 1

    CDS 2 2016

    79
    What is cot⁡A+cosec⁡A−1cot⁡A−cosec⁡A+1\frac{\cot A + \operatorname{cosec} A - 1}{\cot A - \operatorname{cosec} A + 1} equal to ?

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  2. Question 2

    CDS 1 2020

    60
    What is sin⁡θ−cos⁡θ+1sin⁡θ+cos⁡θ−1−sin⁡θ+1cos⁡θ\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} - \frac{\sin\theta + 1}{\cos\theta} equal to ?

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  3. Question 3

    CDS 1 2022

    46
    If x=1−cos⁡θ+sin⁡θ1+sin⁡θx = \frac{1 - \cos\theta + \sin\theta}{1 + \sin\theta} then what is sin⁡θ+cos⁡θ−1cos⁡θ\frac{\sin\theta + \cos\theta - 1}{\cos\theta} equal to?

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  4. Question 4

    CDS 2 2022

    54
    If x=1+sin⁡θcos⁡θx = \frac{1 + \sin\theta}{\cos\theta}, then what is tan⁡θ+sec⁡θ−1tan⁡θ−sec⁡θ+1\frac{\tan\theta + \sec\theta - 1}{\tan\theta - \sec\theta + 1} equal to ?

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  5. Question 5

    CDS 1 2024

    52
    If sin⁡θ−cos⁡θ+1sin⁡θ+cos⁡θ−1=psec⁡θ+qtan⁡θ\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = p\sec\theta + q\tan\theta, where 0<θ<π20 < \theta < \frac{\pi}{2}, then what is p+qp + q equal to ?

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  6. Question 6

    CDS 2 2024

    39
    What is (cos⁡θ−sin⁡θ+1cos⁡θ+sin⁡θ−1)(cot⁡θ−cosec θ)\left(\dfrac{\cos\theta - \sin\theta + 1}{\cos\theta + \sin\theta - 1}\right)(\cot\theta - \mathrm{cosec}\,\theta) equal to ?

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