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Compare or sign-check trig values of acute angles

This type of question, asked 5 times: CDS 2 2018, CDS 2 2020, CDS 1 2021, CDS 2 2024, CDS 1 2025. All 5 questions below, oldest first.

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  1. Question 1

    CDS 2 2018

    62
    If 0<θ<90∘0 < \theta < 90^\circ, 0<ϕ<90∘0 < \phi < 90^\circ and cos⁡θ<cos⁡ϕ\cos \theta < \cos \phi, then which one of the following is correct ?

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  2. Question 2

    CDS 2 2020

    60
    Consider the following statements : 1. The value of cos⁡61∘+sin⁡29∘\cos 61^\circ + \sin 29^\circ cannot exceed 1. 2. The value of tan⁡23∘−cot⁡67∘\tan 23^\circ - \cot 67^\circ is less than 0. Which of the above statements is/are correct ?

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  3. Question 3

    CDS 1 2021

    51
    Consider the following inequalities : 1. sin⁡1∘<cos⁡57∘\sin 1^\circ < \cos 57^\circ 2. cos⁡60∘>sin⁡57∘\cos 60^\circ > \sin 57^\circ Which of the above is/are correct?

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  4. Question 4

    CDS 2 2024

    13
    Consider the following statements : I. tan⁡50∘−cot⁡50∘\tan 50^\circ - \cot 50^\circ is positive II. cot⁡25∘−tan⁡25∘\cot 25^\circ - \tan 25^\circ is negative Which of the statements is/are correct ?

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  5. Question 5

    CDS 1 2025

    24
    Consider the following statements : I. (cosec⁡α−sec⁡α)(\operatorname{cosec} \alpha - \sec \alpha) is always positive in the first quadrant. II. (tan⁡α−cot⁡α)(\tan \alpha - \cot \alpha) is always negative in the first quadrant. Which of the statements given above is/are correct ?

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