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Simplify expressions with (1 + sin +/- cos) or (tan + sec - 1) pairs

This type of question, asked 2 times: 3 Feb 2022 Tier II, 8 Aug 2022 Tier II. All 3 questions below, oldest first.

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  1. Question 1

    3 Feb 2022 Tier II

    64
    The expression cos⁡4θ−sin⁡4θ+2sin⁡2θ+3(cosec⁡θ+cot⁡θ+1)(cosec⁡θ−cot⁡θ+1)−2\frac{\cos^4\theta - \sin^4\theta + 2\sin^2\theta + 3}{(\operatorname{cosec}\theta + \cot\theta + 1)(\operatorname{cosec}\theta - \cot\theta + 1) - 2}, 0∘<θ<90∘0^\circ < \theta < 90^\circ, is equal to:

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  2. Question 2

    3 Feb 2022 Tier II

    66
    The expression (1−sin⁡θ+cos⁡θ)2(1−cos⁡θ)sec⁡3θcosec⁡2θ(sec⁡θ−tan⁡θ)(tan⁡θ+cot⁡θ)\frac{(1 - \sin\theta + \cos\theta)^2(1 - \cos\theta)\sec^3\theta\operatorname{cosec}^2\theta}{(\sec\theta - \tan\theta)(\tan\theta + \cot\theta)}, 0∘<θ<90∘0^\circ < \theta < 90^\circ, is equal to:

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  3. Question 3

    8 Aug 2022 Tier II

    57
    Which of the following is equal to [tan⁡θ+sec⁡θ−1tan⁡θ−sec⁡θ+1]\left[\frac{\tan\theta + \sec\theta - 1}{\tan\theta - \sec\theta + 1}\right] ?

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