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cosec t - sin t and sec t - cos t given: eliminate t

This type of question, asked 3 times: CDS 2 2020, CDS 2 2025, CDS 1 2026. All 4 questions below, oldest first.

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  1. Question 1

    CDS 2 2020

    58
    If cosec⁡θ−sin⁡θ=p3\operatorname{cosec} \theta - \sin \theta = p^3 and sec⁡θ−cos⁡θ=q3\sec \theta - \cos \theta = q^3, then what is the value of tan⁡θ\tan \theta ?

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  2. Question 2

    CDS 2 2020

    46
    If cosec⁡θ−sin⁡θ=m\operatorname{cosec}\theta - \sin\theta = m and sec⁡θ−cos⁡θ=n\sec\theta - \cos\theta = n, then what is m43n23+m23n43m^{\frac{4}{3}} n^{\frac{2}{3}} + m^{\frac{2}{3}} n^{\frac{4}{3}} equal to ?

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  3. Question 3 (case study, 2 parts)

    CDS 2 2025

    Let cosec θ−sin⁡θ=p\mathrm{cosec}\,\theta - \sin\theta = p and sec⁡θ−cos⁡θ=q\sec\theta - \cos\theta = q.
    43
    What is (psin⁡θ+qcos⁡θ)(p\sin\theta + q\cos\theta) equal to?

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    44
    What is p2q2(p2+q2+3)p^2q^2(p^2 + q^2 + 3) equal to?

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  4. Question 4 (case study, 2 parts)

    CDS 1 2026

    For the next two (02) items that follow : cosec⁡θ−sin⁡θ=p3\operatorname{cosec}\theta - \sin\theta = p^3 and sec⁡θ−cos⁡θ=q3\sec\theta - \cos\theta = q^3
    93
    What is tan⁡θ\tan\theta equal to ?

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    94
    What is p4q2+p2q4p^4q^2 + p^2q^4 equal to ?

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