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Equation quadratic in one trig ratio: solve, then evaluate

This type of question, asked 6 times: CDS 2 2021, CDS 1 2022, CDS 1 2023, CDS 2 2023, CDS 1 2025, CDS 2 2025. All 6 questions below, oldest first.

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  1. Question 1

    CDS 2 2021

    52
    If cos⁡2θ−3cos⁡θ+2sin⁡2θ=1\dfrac{\cos^2\theta-3\cos\theta+2}{\sin^2\theta}=1, where 0∘<θ<90∘0^\circ<\theta<90^\circ then what is sin⁡2θ+cos⁡θ\sin^2\theta+\cos\theta equal to ?

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  2. Question 2

    CDS 1 2022

    59
    If tan⁡2θ+3sec⁡θ−9=0\tan^2\theta + 3\sec\theta - 9 = 0, where 0<θ<90∘0 < \theta < 90^\circ, then what is the value of 12cot⁡2θ+3cosec⁡θ12\cot^2\theta + 3\operatorname{cosec}\theta?

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  3. Question 3

    CDS 1 2023

    53
    If 2cos⁡2θ+sin⁡θ−2=02\cos^2\theta + \sin\theta - 2 = 0, 0<θ≤π20 < \theta \le \frac{\pi}{2}, then what is the value of θ\theta ?

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  4. Question 4

    CDS 2 2023

    26
    What is the value of sin⁡θ+cos⁡θ\sin\theta + \cos\theta, if θ\theta satisfies the equation cot⁡2θ−(3+1)cot⁡θ+3=0\cot^2\theta - (\sqrt{3} + 1)\cot\theta + \sqrt{3} = 0; 0<θ<π40 < \theta < \dfrac{\pi}{4} ?

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  5. Question 5

    CDS 1 2025

    28
    If cot⁡2θ−33cot⁡θ+6=0\cot^2\theta - 3\sqrt{3}\cot\theta + 6 = 0, where π6≤θ<π2\frac{\pi}{6} \le \theta < \frac{\pi}{2}, then what is a value of sin⁡θ+cos⁡2θ\sin\theta + \cos 2\theta ?

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  6. Question 6

    CDS 2 2025

    33
    If 2tan⁡θ=sec⁡2θ−22\tan\theta = \sec^2\theta - 2, where 0<θ<π/20 < \theta < \pi/2, then what is cot⁡θ\cot\theta equal to?

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