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x + 1/x >= 2: bound sin x + cosec x, cos x + sec x and similar

This type of question, asked 5 times: CDS 2 2017, CDS 2 2018, CDS 2 2021, CDS 1 2022, CDS 1 2024. All 5 questions below, oldest first.

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  1. Question 1

    CDS 2 2017

    47
    If 0<x<π20 < x < \frac{\pi}{2}, then (sin⁡x+cosec⁡x)(\sin x + \operatorname{cosec} x) is

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  2. Question 2

    CDS 2 2018

    67
    Consider the following statements : 1. cos⁡θ+sec⁡θ\cos \theta + \sec \theta can never be equal to 1.5. 2. sec⁡2θ+cosec⁡2θ\sec^2 \theta + \operatorname{cosec}^2 \theta can never be less than 4. Which of the statements given above is/are correct ?

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  3. Question 3

    CDS 2 2021

    46
    What is the minimum value of cos⁡3θ+sec⁡3θ\cos^3\theta+\sec^3\theta where 0∘≤θ<90∘0^\circ\le\theta<90^\circ ?

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  4. Question 4

    CDS 1 2022

    50
    If y=cos⁡2x+sec⁡2xy = \cos^2 x + \sec^2 x, where 0≤x<π20 \le x < \frac{\pi}{2}, then which one of the following is correct?

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  5. Question 5

    CDS 1 2024

    34
    What is the minimum value of sin⁡2A+5sin⁡A+1sin⁡A\frac{\sin^2 A + 5\sin A + 1}{\sin A} for 0<A≤π20 < A \le \frac{\pi}{2} ?

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