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p and q defined from sec +/- tan and cosec +/- cot: evaluate a combination

This type of question, asked 3 times: CDS 1 2021, CDS 1 2025, CDS 2 2026. All 3 questions below, oldest first.

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  1. Question 1

    CDS 1 2021

    52
    If p=sec⁡θ−tan⁡θp = \sec\theta - \tan\theta and q=cosec⁡θ+cot⁡θq = \operatorname{cosec}\theta + \cot\theta, then what is p+q(p−1)p + q(p - 1) equal to?

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  2. Question 2

    CDS 1 2025

    23
    If p=1cosec⁡θ+cot⁡θp = \dfrac{1}{\operatorname{cosec} \theta + \cot \theta} and q=cosec⁡θq = \operatorname{cosec} \theta, then what is p2−2pqp^2 - 2pq equal to ?

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  3. Question 3

    CDS 2 2026

    50
    If p=sec⁡θ+tan⁡θp = \sec\theta + \tan\theta and q=cosec⁡θ−cot⁡θq = \operatorname{cosec}\theta - \cot\theta, then what is (p−q−pq)(p - q - pq) equal to ?

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