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cos t/(1 - sin t) + cos t/(1 + sin t) = k: solve for t

This type of question, asked 3 times: CDS 2 2018, CDS 1 2022, CDS 1 2026. All 3 questions below, oldest first.

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  1. Question 1

    CDS 2 2018

    65
    Consider the following statements : 1. If cos⁡θ1−sin⁡θ+cos⁡θ1+sin⁡θ=4\frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} = 4, where 0<θ<90∘0 < \theta < 90^\circ, then θ=60∘\theta = 60^\circ. 2. If 3tan⁡θ+cot⁡θ=5cosec⁡θ3 \tan \theta + \cot \theta = 5 \operatorname{cosec} \theta, where 0<θ<90∘0 < \theta < 90^\circ, then θ=60∘\theta = 60^\circ. Which of the statements given above is/are correct ?

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  2. Question 2

    CDS 1 2022

    60
    If cos⁡θcosec⁡θ+1+cos⁡θcosec⁡θ−1=2\dfrac{\cos\theta}{\operatorname{cosec}\theta + 1} + \dfrac{\cos\theta}{\operatorname{cosec}\theta - 1} = 2 where 0<θ<90∘0 < \theta < 90^\circ, then what is the value of sin⁡4θ+cos⁡4θ\sin^4\theta + \cos^4\theta?

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  3. Question 3

    CDS 1 2026

    41
    If cos⁡θ1−sin⁡θ+cos⁡θ1+sin⁡θ=4\frac{\cos\theta}{1 - \sin\theta} + \frac{\cos\theta}{1 + \sin\theta} = 4, then which one of the following is a value of (tan⁡2θ+cot⁡2θ)(\tan^2\theta + \cot^2\theta) ?

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